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Properties

Label 2-510e2-1.1-c1-0-39
Degree $2$
Conductor $260100$
Sign $-1$
Analytic cond. $2076.90$
Root an. cond. $45.5731$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·7-s − 3·11-s − 4·13-s − 5·19-s + 2·23-s − 29-s − 2·37-s + 11·41-s + 2·43-s − 2·47-s + 9·49-s + 2·53-s + 5·59-s − 13·61-s − 8·67-s − 15·71-s − 4·73-s + 12·77-s − 9·79-s + 14·83-s − 11·89-s + 16·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯
L(s)  = 1  − 1.51·7-s − 0.904·11-s − 1.10·13-s − 1.14·19-s + 0.417·23-s − 0.185·29-s − 0.328·37-s + 1.71·41-s + 0.304·43-s − 0.291·47-s + 9/7·49-s + 0.274·53-s + 0.650·59-s − 1.66·61-s − 0.977·67-s − 1.78·71-s − 0.468·73-s + 1.36·77-s − 1.01·79-s + 1.53·83-s − 1.16·89-s + 1.67·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(260100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(2076.90\)
Root analytic conductor: \(45.5731\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 260100,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 5 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.94378930404716, −12.68549863725895, −12.23117947429476, −11.85090236792960, −10.99745753556730, −10.72163595490704, −10.27896065761046, −9.744640185988263, −9.479957569487812, −8.894156622897609, −8.489384347372212, −7.778055574672544, −7.319112196649795, −7.030128078086885, −6.369897624883695, −5.917852679779587, −5.562090000793158, −4.790938505930715, −4.387261951183133, −3.835721135571988, −3.027086516893024, −2.794779892282168, −2.291991252431583, −1.500641454850062, −0.4896327283330937, 0, 0.4896327283330937, 1.500641454850062, 2.291991252431583, 2.794779892282168, 3.027086516893024, 3.835721135571988, 4.387261951183133, 4.790938505930715, 5.562090000793158, 5.917852679779587, 6.369897624883695, 7.030128078086885, 7.319112196649795, 7.778055574672544, 8.489384347372212, 8.894156622897609, 9.479957569487812, 9.744640185988263, 10.27896065761046, 10.72163595490704, 10.99745753556730, 11.85090236792960, 12.23117947429476, 12.68549863725895, 12.94378930404716

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