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Properties

Label 4-10e2-1.1-c25e2-0-0
Degree $4$
Conductor $100$
Sign $1$
Analytic cond. $1568.13$
Root an. cond. $6.29282$
Motivic weight $25$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.19e3·2-s − 4.38e5·3-s + 5.03e7·4-s + 4.88e8·5-s + 3.59e9·6-s − 3.40e10·7-s − 2.74e11·8-s + 4.61e11·9-s − 4.00e12·10-s + 1.93e13·11-s − 2.20e13·12-s − 1.27e14·13-s + 2.78e14·14-s − 2.14e14·15-s + 1.40e15·16-s − 2.99e15·17-s − 3.78e15·18-s + 1.35e13·19-s + 2.45e16·20-s + 1.49e16·21-s − 1.58e17·22-s − 1.76e17·23-s + 1.20e17·24-s + 1.78e17·25-s + 1.04e18·26-s − 6.92e17·27-s − 1.71e18·28-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.476·3-s + 3/2·4-s + 0.894·5-s + 0.673·6-s − 0.928·7-s − 1.41·8-s + 0.545·9-s − 1.26·10-s + 1.85·11-s − 0.714·12-s − 1.52·13-s + 1.31·14-s − 0.426·15-s + 5/4·16-s − 1.24·17-s − 0.771·18-s + 0.00140·19-s + 1.34·20-s + 0.442·21-s − 2.62·22-s − 1.68·23-s + 0.673·24-s + 3/5·25-s + 2.15·26-s − 0.887·27-s − 1.39·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+25/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(1568.13\)
Root analytic conductor: \(6.29282\)
Motivic weight: \(25\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 100,\ (\ :25/2, 25/2),\ 1)\)

Particular Values

\(L(13)\) \(\approx\) \(0.7521744680\)
\(L(\frac12)\) \(\approx\) \(0.7521744680\)
\(L(\frac{27}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_1$ \( ( 1 + p^{12} T )^{2} \)
5$C_1$ \( ( 1 - p^{12} T )^{2} \)
good3$D_{4}$ \( 1 + 16244 p^{3} T - 369862282 p^{6} T^{2} + 16244 p^{28} T^{3} + p^{50} T^{4} \)
7$D_{4}$ \( 1 + 4858370948 p T + 1231557749974659438 p^{4} T^{2} + 4858370948 p^{26} T^{3} + p^{50} T^{4} \)
11$D_{4}$ \( 1 - 159873411504 p^{2} T + \)\(25\!\cdots\!06\)\( p T^{2} - 159873411504 p^{27} T^{3} + p^{50} T^{4} \)
13$D_{4}$ \( 1 + 127796048033588 T + \)\(10\!\cdots\!94\)\( p T^{2} + 127796048033588 p^{25} T^{3} + p^{50} T^{4} \)
17$D_{4}$ \( 1 + 2998857935578476 T + \)\(23\!\cdots\!74\)\( p T^{2} + 2998857935578476 p^{25} T^{3} + p^{50} T^{4} \)
19$D_{4}$ \( 1 - 13534782395320 T + \)\(81\!\cdots\!42\)\( p T^{2} - 13534782395320 p^{25} T^{3} + p^{50} T^{4} \)
23$D_{4}$ \( 1 + 7686979164461916 p T + \)\(48\!\cdots\!98\)\( p^{2} T^{2} + 7686979164461916 p^{26} T^{3} + p^{50} T^{4} \)
29$D_{4}$ \( 1 + 1676097487309033380 T + \)\(69\!\cdots\!98\)\( T^{2} + 1676097487309033380 p^{25} T^{3} + p^{50} T^{4} \)
31$D_{4}$ \( 1 - 7125865156346376664 T + \)\(43\!\cdots\!26\)\( T^{2} - 7125865156346376664 p^{25} T^{3} + p^{50} T^{4} \)
37$D_{4}$ \( 1 - 84009823521808944364 T + \)\(48\!\cdots\!38\)\( T^{2} - 84009823521808944364 p^{25} T^{3} + p^{50} T^{4} \)
41$D_{4}$ \( 1 + \)\(27\!\cdots\!56\)\( T + \)\(47\!\cdots\!86\)\( T^{2} + \)\(27\!\cdots\!56\)\( p^{25} T^{3} + p^{50} T^{4} \)
43$D_{4}$ \( 1 - \)\(42\!\cdots\!32\)\( T + \)\(12\!\cdots\!42\)\( T^{2} - \)\(42\!\cdots\!32\)\( p^{25} T^{3} + p^{50} T^{4} \)
47$D_{4}$ \( 1 - \)\(11\!\cdots\!44\)\( T + \)\(13\!\cdots\!98\)\( T^{2} - \)\(11\!\cdots\!44\)\( p^{25} T^{3} + p^{50} T^{4} \)
53$D_{4}$ \( 1 - \)\(41\!\cdots\!52\)\( T + \)\(29\!\cdots\!62\)\( T^{2} - \)\(41\!\cdots\!52\)\( p^{25} T^{3} + p^{50} T^{4} \)
59$D_{4}$ \( 1 + \)\(37\!\cdots\!60\)\( T - \)\(67\!\cdots\!02\)\( T^{2} + \)\(37\!\cdots\!60\)\( p^{25} T^{3} + p^{50} T^{4} \)
61$D_{4}$ \( 1 - \)\(10\!\cdots\!24\)\( T + \)\(85\!\cdots\!46\)\( T^{2} - \)\(10\!\cdots\!24\)\( p^{25} T^{3} + p^{50} T^{4} \)
67$D_{4}$ \( 1 + \)\(11\!\cdots\!56\)\( T + \)\(10\!\cdots\!98\)\( T^{2} + \)\(11\!\cdots\!56\)\( p^{25} T^{3} + p^{50} T^{4} \)
71$D_{4}$ \( 1 - \)\(12\!\cdots\!24\)\( T + \)\(15\!\cdots\!46\)\( T^{2} - \)\(12\!\cdots\!24\)\( p^{25} T^{3} + p^{50} T^{4} \)
73$D_{4}$ \( 1 - \)\(15\!\cdots\!72\)\( T - \)\(29\!\cdots\!18\)\( T^{2} - \)\(15\!\cdots\!72\)\( p^{25} T^{3} + p^{50} T^{4} \)
79$D_{4}$ \( 1 - \)\(82\!\cdots\!40\)\( T + \)\(68\!\cdots\!98\)\( T^{2} - \)\(82\!\cdots\!40\)\( p^{25} T^{3} + p^{50} T^{4} \)
83$D_{4}$ \( 1 - \)\(10\!\cdots\!52\)\( T + \)\(58\!\cdots\!62\)\( T^{2} - \)\(10\!\cdots\!52\)\( p^{25} T^{3} + p^{50} T^{4} \)
89$D_{4}$ \( 1 - \)\(50\!\cdots\!80\)\( T + \)\(16\!\cdots\!98\)\( T^{2} - \)\(50\!\cdots\!80\)\( p^{25} T^{3} + p^{50} T^{4} \)
97$D_{4}$ \( 1 - \)\(23\!\cdots\!64\)\( T + \)\(86\!\cdots\!38\)\( T^{2} - \)\(23\!\cdots\!64\)\( p^{25} T^{3} + p^{50} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.35560538302981314246190502016, −14.69647531187002383560856252262, −13.76532216672891658499231437872, −12.92441769455216972812422156873, −11.94776194834140316010980880996, −11.72639571803384056535805664769, −10.55399977179044430738966346455, −9.921451582136874571805547122891, −9.424417231017288195976173630680, −9.086705326735111049542067541225, −7.83649584707832287144573750568, −7.02974572250362034723139795297, −6.25875615171753723974885942184, −6.13486572165067192589169489293, −4.64732143207354616496996889314, −3.73951058896370321078078363188, −2.42613107860636242910358284816, −2.03057451311127191073250853394, −1.10375898002819118815462173163, −0.35983540391047310159146841172, 0.35983540391047310159146841172, 1.10375898002819118815462173163, 2.03057451311127191073250853394, 2.42613107860636242910358284816, 3.73951058896370321078078363188, 4.64732143207354616496996889314, 6.13486572165067192589169489293, 6.25875615171753723974885942184, 7.02974572250362034723139795297, 7.83649584707832287144573750568, 9.086705326735111049542067541225, 9.424417231017288195976173630680, 9.921451582136874571805547122891, 10.55399977179044430738966346455, 11.72639571803384056535805664769, 11.94776194834140316010980880996, 12.92441769455216972812422156873, 13.76532216672891658499231437872, 14.69647531187002383560856252262, 15.35560538302981314246190502016

Graph of the $Z$-function along the critical line