Properties

Label 2-10e2-1.1-c3-0-4
Degree $2$
Conductor $100$
Sign $-1$
Analytic cond. $5.90019$
Root an. cond. $2.42903$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 26·7-s − 26·9-s + 45·11-s − 44·13-s − 117·17-s − 91·19-s + 26·21-s + 18·23-s + 53·27-s + 144·29-s + 26·31-s − 45·33-s + 214·37-s + 44·39-s − 459·41-s + 460·43-s + 468·47-s + 333·49-s + 117·51-s − 558·53-s + 91·57-s − 72·59-s − 118·61-s + 676·63-s − 251·67-s − 18·69-s + ⋯
L(s)  = 1  − 0.192·3-s − 1.40·7-s − 0.962·9-s + 1.23·11-s − 0.938·13-s − 1.66·17-s − 1.09·19-s + 0.270·21-s + 0.163·23-s + 0.377·27-s + 0.922·29-s + 0.150·31-s − 0.237·33-s + 0.950·37-s + 0.180·39-s − 1.74·41-s + 1.63·43-s + 1.45·47-s + 0.970·49-s + 0.321·51-s − 1.44·53-s + 0.211·57-s − 0.158·59-s − 0.247·61-s + 1.35·63-s − 0.457·67-s − 0.0314·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(100\)    =    \(2^{2} \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(5.90019\)
Root analytic conductor: \(2.42903\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{100} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 100,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + T + p^{3} T^{2} \)
7 \( 1 + 26 T + p^{3} T^{2} \)
11 \( 1 - 45 T + p^{3} T^{2} \)
13 \( 1 + 44 T + p^{3} T^{2} \)
17 \( 1 + 117 T + p^{3} T^{2} \)
19 \( 1 + 91 T + p^{3} T^{2} \)
23 \( 1 - 18 T + p^{3} T^{2} \)
29 \( 1 - 144 T + p^{3} T^{2} \)
31 \( 1 - 26 T + p^{3} T^{2} \)
37 \( 1 - 214 T + p^{3} T^{2} \)
41 \( 1 + 459 T + p^{3} T^{2} \)
43 \( 1 - 460 T + p^{3} T^{2} \)
47 \( 1 - 468 T + p^{3} T^{2} \)
53 \( 1 + 558 T + p^{3} T^{2} \)
59 \( 1 + 72 T + p^{3} T^{2} \)
61 \( 1 + 118 T + p^{3} T^{2} \)
67 \( 1 + 251 T + p^{3} T^{2} \)
71 \( 1 - 108 T + p^{3} T^{2} \)
73 \( 1 + 299 T + p^{3} T^{2} \)
79 \( 1 + 898 T + p^{3} T^{2} \)
83 \( 1 + 927 T + p^{3} T^{2} \)
89 \( 1 - 351 T + p^{3} T^{2} \)
97 \( 1 + 386 T + p^{3} T^{2} \)
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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

−12.78353473466010716231752512484, −11.89211348679333791306236286048, −10.75701834579732585184718824250, −9.482018432090836976601182730539, −8.689021176182826199960860675175, −6.83955769501248664216119724456, −6.14124627160999848738309591102, −4.32822212979424740304603494496, −2.70294939533323920262734245917, 0, 2.70294939533323920262734245917, 4.32822212979424740304603494496, 6.14124627160999848738309591102, 6.83955769501248664216119724456, 8.689021176182826199960860675175, 9.482018432090836976601182730539, 10.75701834579732585184718824250, 11.89211348679333791306236286048, 12.78353473466010716231752512484

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