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Properties

Label 2-50e2-2500.11-c0-0-0
Degree $2$
Conductor $2500$
Sign $0.997 + 0.0703i$
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  + (0.988 − 0.150i)2-s + (0.954 − 0.297i)4-s + (0.492 + 0.870i)5-s + (0.899 − 0.437i)8-s + (0.656 − 0.754i)9-s + (0.617 + 0.786i)10-s + (−0.268 − 0.447i)13-s + (0.823 − 0.567i)16-s + (−0.594 + 0.839i)17-s + (0.535 − 0.844i)18-s + (0.728 + 0.684i)20-s + (−0.514 + 0.857i)25-s + (−0.332 − 0.401i)26-s + (−1.52 − 1.29i)29-s + (0.728 − 0.684i)32-s + ⋯
L(s)  = 1  + (0.988 − 0.150i)2-s + (0.954 − 0.297i)4-s + (0.492 + 0.870i)5-s + (0.899 − 0.437i)8-s + (0.656 − 0.754i)9-s + (0.617 + 0.786i)10-s + (−0.268 − 0.447i)13-s + (0.823 − 0.567i)16-s + (−0.594 + 0.839i)17-s + (0.535 − 0.844i)18-s + (0.728 + 0.684i)20-s + (−0.514 + 0.857i)25-s + (−0.332 − 0.401i)26-s + (−1.52 − 1.29i)29-s + (0.728 − 0.684i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2500\)    =    \(2^{2} \cdot 5^{4}\)
Sign: $0.997 + 0.0703i$
Analytic conductor: \(1.24766\)
Root analytic conductor: \(1.11698\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2500} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2500,\ (\ :0),\ 0.997 + 0.0703i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.527914465\)
\(L(\frac12)\) \(\approx\) \(2.527914465\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.988 + 0.150i)T \)
5 \( 1 + (-0.492 - 0.870i)T \)
good3 \( 1 + (-0.656 + 0.754i)T^{2} \)
7 \( 1 + (0.187 + 0.982i)T^{2} \)
11 \( 1 + (-0.577 - 0.816i)T^{2} \)
13 \( 1 + (0.268 + 0.447i)T + (-0.470 + 0.882i)T^{2} \)
17 \( 1 + (0.594 - 0.839i)T + (-0.332 - 0.942i)T^{2} \)
19 \( 1 + (-0.920 + 0.391i)T^{2} \)
23 \( 1 + (0.379 - 0.925i)T^{2} \)
29 \( 1 + (1.52 + 1.29i)T + (0.162 + 0.986i)T^{2} \)
31 \( 1 + (0.332 + 0.942i)T^{2} \)
37 \( 1 + (-0.981 - 0.751i)T + (0.260 + 0.965i)T^{2} \)
41 \( 1 + (1.77 - 0.652i)T + (0.762 - 0.647i)T^{2} \)
43 \( 1 + (0.637 + 0.770i)T^{2} \)
47 \( 1 + (0.556 - 0.830i)T^{2} \)
53 \( 1 + (-0.606 - 1.59i)T + (-0.745 + 0.666i)T^{2} \)
59 \( 1 + (0.910 + 0.414i)T^{2} \)
61 \( 1 + (1.26 + 0.465i)T + (0.762 + 0.647i)T^{2} \)
67 \( 1 + (0.711 - 0.702i)T^{2} \)
71 \( 1 + (-0.938 + 0.344i)T^{2} \)
73 \( 1 + (-1.78 - 0.180i)T + (0.979 + 0.199i)T^{2} \)
79 \( 1 + (-0.656 + 0.754i)T^{2} \)
83 \( 1 + (0.514 - 0.857i)T^{2} \)
89 \( 1 + (-1.06 - 0.947i)T + (0.112 + 0.993i)T^{2} \)
97 \( 1 + (1.26 - 0.783i)T + (0.448 - 0.893i)T^{2} \)
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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

−9.426098046083380513573806401296, −8.081985583479468560720802763812, −7.32122652261691633556856739023, −6.51736167789708712650119433599, −6.12232821842786756386432971603, −5.19782992867647482710499740419, −4.14590180157420438027989290690, −3.49902494806640224424504256001, −2.52944988393663734718677550128, −1.59531583269045393493323939513, 1.63066265494824015058570201051, 2.33391227833640534696180293720, 3.63509463563814613153495003764, 4.58078724620433252170199644102, 5.05876141022346509709482341513, 5.77015933552895182962507791669, 6.82424515619557509412857166362, 7.35392541342797282625076942428, 8.233345258645179668235197100726, 9.128331970349584522098046471441

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