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Properties

Label 2-50e2-2500.1071-c0-0-0
Degree $2$
Conductor $2500$
Sign $-0.629 + 0.776i$
Analytic cond. $1.24766$
Root an. cond. $1.11698$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.693 − 0.720i)2-s + (−0.0376 − 0.999i)4-s + (0.793 − 0.607i)5-s + (−0.745 − 0.666i)8-s + (−0.778 − 0.627i)9-s + (0.112 − 0.993i)10-s + (0.0848 + 0.314i)13-s + (−0.997 + 0.0753i)16-s + (0.322 − 0.410i)17-s + (−0.992 + 0.125i)18-s + (−0.637 − 0.770i)20-s + (0.260 − 0.965i)25-s + (0.285 + 0.156i)26-s + (0.0667 − 0.756i)29-s + (−0.637 + 0.770i)32-s + ⋯
L(s)  = 1  + (0.693 − 0.720i)2-s + (−0.0376 − 0.999i)4-s + (0.793 − 0.607i)5-s + (−0.745 − 0.666i)8-s + (−0.778 − 0.627i)9-s + (0.112 − 0.993i)10-s + (0.0848 + 0.314i)13-s + (−0.997 + 0.0753i)16-s + (0.322 − 0.410i)17-s + (−0.992 + 0.125i)18-s + (−0.637 − 0.770i)20-s + (0.260 − 0.965i)25-s + (0.285 + 0.156i)26-s + (0.0667 − 0.756i)29-s + (−0.637 + 0.770i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $-0.629 + 0.776i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (1071, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ -0.629 + 0.776i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.768716308\)
\(L(\frac12)\) \(\approx\) \(1.768716308\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.693 + 0.720i)T \)
5 \( 1 + (-0.793 + 0.607i)T \)
good3 \( 1 + (0.778 + 0.627i)T^{2} \)
7 \( 1 + (-0.535 + 0.844i)T^{2} \)
11 \( 1 + (-0.617 - 0.786i)T^{2} \)
13 \( 1 + (-0.0848 - 0.314i)T + (-0.863 + 0.503i)T^{2} \)
17 \( 1 + (-0.322 + 0.410i)T + (-0.236 - 0.971i)T^{2} \)
19 \( 1 + (-0.998 + 0.0502i)T^{2} \)
23 \( 1 + (0.514 + 0.857i)T^{2} \)
29 \( 1 + (-0.0667 + 0.756i)T + (-0.984 - 0.175i)T^{2} \)
31 \( 1 + (0.236 + 0.971i)T^{2} \)
37 \( 1 + (-0.172 + 0.146i)T + (0.162 - 0.986i)T^{2} \)
41 \( 1 + (1.26 - 1.38i)T + (-0.0878 - 0.996i)T^{2} \)
43 \( 1 + (-0.876 - 0.481i)T^{2} \)
47 \( 1 + (-0.492 - 0.870i)T^{2} \)
53 \( 1 + (-1.29 - 0.197i)T + (0.954 + 0.297i)T^{2} \)
59 \( 1 + (0.332 + 0.942i)T^{2} \)
61 \( 1 + (-0.384 - 0.420i)T + (-0.0878 + 0.996i)T^{2} \)
67 \( 1 + (0.470 + 0.882i)T^{2} \)
71 \( 1 + (0.675 - 0.737i)T^{2} \)
73 \( 1 + (0.0187 - 1.49i)T + (-0.999 - 0.0251i)T^{2} \)
79 \( 1 + (0.778 + 0.627i)T^{2} \)
83 \( 1 + (-0.260 + 0.965i)T^{2} \)
89 \( 1 + (0.898 - 0.279i)T + (0.823 - 0.567i)T^{2} \)
97 \( 1 + (-1.25 + 1.44i)T + (-0.137 - 0.990i)T^{2} \)
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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

−8.937578196853097615243806044421, −8.419684291230679224107893458881, −7.02955218797295608011857968406, −6.19535809194374374098375415559, −5.62850503250139680797277859296, −4.88617100354159922323054388419, −3.98454215642036850568991599064, −2.99613789846421101620135517171, −2.12670287818967435412116347012, −0.941623602071538857466862987841, 2.02677585822234168383570911551, 2.93206786366194521051409006793, 3.69905762504625492742038953681, 4.95531527882495981025909843813, 5.54440448465257569403140338064, 6.16489429756434333378776830313, 6.98245685189090879682879285461, 7.68449603393771555079351534400, 8.526399153954050044955240401060, 9.113829833073481042519454403692

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