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Properties

Label 4-15e2-1.1-c19e2-0-0
Degree $4$
Conductor $225$
Sign $1$
Analytic cond. $1178.03$
Root an. cond. $5.85854$
Motivic weight $19$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 152·2-s − 3.93e4·3-s − 8.64e5·4-s + 3.90e6·5-s − 5.98e6·6-s − 7.36e5·7-s − 1.86e8·8-s + 1.16e9·9-s + 5.93e8·10-s + 5.36e9·11-s + 3.40e10·12-s + 5.17e10·13-s − 1.11e8·14-s − 1.53e11·15-s + 4.75e11·16-s − 2.50e11·17-s + 1.76e11·18-s − 1.11e12·19-s − 3.37e12·20-s + 2.89e10·21-s + 8.16e11·22-s − 5.92e12·23-s + 7.34e12·24-s + 1.14e13·25-s + 7.86e12·26-s − 3.05e13·27-s + 6.36e11·28-s + ⋯
L(s)  = 1  + 0.209·2-s − 1.15·3-s − 1.64·4-s + 0.894·5-s − 0.242·6-s − 0.00689·7-s − 0.491·8-s + 9-s + 0.187·10-s + 0.686·11-s + 1.90·12-s + 1.35·13-s − 0.00144·14-s − 1.03·15-s + 1.72·16-s − 0.511·17-s + 0.209·18-s − 0.791·19-s − 1.47·20-s + 0.00796·21-s + 0.144·22-s − 0.686·23-s + 0.567·24-s + 3/5·25-s + 0.283·26-s − 0.769·27-s + 0.0113·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(225\)    =    \(3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(1178.03\)
Root analytic conductor: \(5.85854\)
Motivic weight: \(19\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 225,\ (\ :19/2, 19/2),\ 1)\)

Particular Values

\(L(10)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{21}{2})\) 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)$
bad3$C_1$ \( ( 1 + p^{9} T )^{2} \)
5$C_1$ \( ( 1 - p^{9} T )^{2} \)
good2$D_{4}$ \( 1 - 19 p^{3} T + 6931 p^{7} T^{2} - 19 p^{22} T^{3} + p^{38} T^{4} \)
7$D_{4}$ \( 1 + 15024 p^{2} T + 288871244163854 p^{2} T^{2} + 15024 p^{21} T^{3} + p^{38} T^{4} \)
11$D_{4}$ \( 1 - 5369697128 T + \)\(12\!\cdots\!34\)\( T^{2} - 5369697128 p^{19} T^{3} + p^{38} T^{4} \)
13$D_{4}$ \( 1 - 51722759612 T + \)\(17\!\cdots\!22\)\( p T^{2} - 51722759612 p^{19} T^{3} + p^{38} T^{4} \)
17$D_{4}$ \( 1 + 14719974956 p T + \)\(11\!\cdots\!38\)\( p^{2} T^{2} + 14719974956 p^{20} T^{3} + p^{38} T^{4} \)
19$D_{4}$ \( 1 + 1113139984504 T + \)\(13\!\cdots\!38\)\( T^{2} + 1113139984504 p^{19} T^{3} + p^{38} T^{4} \)
23$D_{4}$ \( 1 + 5929365574992 T + \)\(81\!\cdots\!54\)\( T^{2} + 5929365574992 p^{19} T^{3} + p^{38} T^{4} \)
29$D_{4}$ \( 1 + 141119811247028 T + \)\(12\!\cdots\!38\)\( T^{2} + 141119811247028 p^{19} T^{3} + p^{38} T^{4} \)
31$D_{4}$ \( 1 - 37508751850032 T + \)\(13\!\cdots\!42\)\( T^{2} - 37508751850032 p^{19} T^{3} + p^{38} T^{4} \)
37$D_{4}$ \( 1 + 1368048785415476 T + \)\(16\!\cdots\!66\)\( T^{2} + 1368048785415476 p^{19} T^{3} + p^{38} T^{4} \)
41$D_{4}$ \( 1 + 4174938760306988 T + \)\(12\!\cdots\!22\)\( T^{2} + 4174938760306988 p^{19} T^{3} + p^{38} T^{4} \)
43$D_{4}$ \( 1 + 7080534797769640 T + \)\(30\!\cdots\!90\)\( T^{2} + 7080534797769640 p^{19} T^{3} + p^{38} T^{4} \)
47$D_{4}$ \( 1 + 239738716958080 T + \)\(14\!\cdots\!90\)\( T^{2} + 239738716958080 p^{19} T^{3} + p^{38} T^{4} \)
53$D_{4}$ \( 1 + 29662427886344452 T + \)\(12\!\cdots\!74\)\( T^{2} + 29662427886344452 p^{19} T^{3} + p^{38} T^{4} \)
59$D_{4}$ \( 1 - 55456595574036584 T + \)\(81\!\cdots\!58\)\( T^{2} - 55456595574036584 p^{19} T^{3} + p^{38} T^{4} \)
61$D_{4}$ \( 1 - 98673648121778540 T + \)\(14\!\cdots\!78\)\( T^{2} - 98673648121778540 p^{19} T^{3} + p^{38} T^{4} \)
67$D_{4}$ \( 1 - 546332988026030088 T + \)\(16\!\cdots\!42\)\( T^{2} - 546332988026030088 p^{19} T^{3} + p^{38} T^{4} \)
71$D_{4}$ \( 1 - 385389801423355024 T + \)\(32\!\cdots\!06\)\( T^{2} - 385389801423355024 p^{19} T^{3} + p^{38} T^{4} \)
73$D_{4}$ \( 1 - 117641357804062868 T + \)\(50\!\cdots\!54\)\( T^{2} - 117641357804062868 p^{19} T^{3} + p^{38} T^{4} \)
79$D_{4}$ \( 1 + 1761854290669138800 T + \)\(30\!\cdots\!38\)\( T^{2} + 1761854290669138800 p^{19} T^{3} + p^{38} T^{4} \)
83$D_{4}$ \( 1 + 515530924759284216 T + \)\(49\!\cdots\!62\)\( T^{2} + 515530924759284216 p^{19} T^{3} + p^{38} T^{4} \)
89$D_{4}$ \( 1 + 5056356550608812364 T + \)\(28\!\cdots\!38\)\( T^{2} + 5056356550608812364 p^{19} T^{3} + p^{38} T^{4} \)
97$D_{4}$ \( 1 + 16093051291454986172 T + \)\(17\!\cdots\!62\)\( T^{2} + 16093051291454986172 p^{19} T^{3} + p^{38} T^{4} \)
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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

−14.08660212083403221148850325198, −13.71863843205280245955510051056, −12.94233815165827340245186923321, −12.75379678821472610825769795536, −11.59482464417535036428390156310, −11.05542151333181395438340075711, −9.970956006194830176607338782335, −9.777308558579070196697207373503, −8.670342848544139146931256782602, −8.409213975603382999804158376947, −6.72564568323460323483379860201, −6.33637301882601181680864099660, −5.30772633068658780232206467612, −5.11172431517598254424082183463, −3.94890819227362916676855355104, −3.62424924811050982498602677879, −1.77862757923474254345497021518, −1.31841690715767650366829797885, 0, 0, 1.31841690715767650366829797885, 1.77862757923474254345497021518, 3.62424924811050982498602677879, 3.94890819227362916676855355104, 5.11172431517598254424082183463, 5.30772633068658780232206467612, 6.33637301882601181680864099660, 6.72564568323460323483379860201, 8.409213975603382999804158376947, 8.670342848544139146931256782602, 9.777308558579070196697207373503, 9.970956006194830176607338782335, 11.05542151333181395438340075711, 11.59482464417535036428390156310, 12.75379678821472610825769795536, 12.94233815165827340245186923321, 13.71863843205280245955510051056, 14.08660212083403221148850325198

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