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Properties

Label 2-1250-5.4-c1-0-15
Degree $2$
Conductor $1250$
Sign $-i$
Analytic cond. $9.98130$
Root an. cond. $3.15931$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 0.381i·3-s − 4-s + 0.381·6-s + 3i·7-s − i·8-s + 2.85·9-s + 4.23·11-s + 0.381i·12-s + i·13-s − 3·14-s + 16-s − 1.14i·17-s + 2.85i·18-s − 5.85·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.220i·3-s − 0.5·4-s + 0.155·6-s + 1.13i·7-s − 0.353i·8-s + 0.951·9-s + 1.27·11-s + 0.110i·12-s + 0.277i·13-s − 0.801·14-s + 0.250·16-s − 0.277i·17-s + 0.672i·18-s − 1.34·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1250\)    =    \(2 \cdot 5^{4}\)
Sign: $-i$
Analytic conductor: \(9.98130\)
Root analytic conductor: \(3.15931\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1250} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1250,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.722037243\)
\(L(\frac12)\) \(\approx\) \(1.722037243\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
good3 \( 1 + 0.381iT - 3T^{2} \)
7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 - 4.23T + 11T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + 1.14iT - 17T^{2} \)
19 \( 1 + 5.85T + 19T^{2} \)
23 \( 1 + 1.76iT - 23T^{2} \)
29 \( 1 - 9.47T + 29T^{2} \)
31 \( 1 + 0.236T + 31T^{2} \)
37 \( 1 - 8.32iT - 37T^{2} \)
41 \( 1 - 1.47T + 41T^{2} \)
43 \( 1 + 6.23iT - 43T^{2} \)
47 \( 1 - 11.9iT - 47T^{2} \)
53 \( 1 - 10.4iT - 53T^{2} \)
59 \( 1 - 4.47T + 59T^{2} \)
61 \( 1 + 8.85T + 61T^{2} \)
67 \( 1 - 10.2iT - 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 - 7.70iT - 73T^{2} \)
79 \( 1 - 7.23T + 79T^{2} \)
83 \( 1 + 4.52iT - 83T^{2} \)
89 \( 1 + 4.47T + 89T^{2} \)
97 \( 1 + 9.56iT - 97T^{2} \)
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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

−9.684861544738837969581865009524, −8.873882486207604783752026943169, −8.426390028753851260792260410507, −7.29399573101097526938258364320, −6.48742570445656358083308216480, −6.06832231843345329201761513885, −4.73586548805721697813321347915, −4.15269854597057949304626604280, −2.66779935051917070893247625101, −1.34650779124563106976574123055, 0.859212574906169052191574286132, 1.95776957835675333042976725576, 3.55823356814939750772689012486, 4.10728643025840240001546436719, 4.84112942939226211244088273030, 6.31484923910832774232050583647, 6.96627497438097932200129665552, 7.961716926358658183648969566382, 8.880163212492177630404272659946, 9.684904124025644434922018586049

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