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Properties

Label 2-325-13.3-c1-0-5
Degree $2$
Conductor $325$
Sign $-0.252 + 0.967i$
Analytic cond. $2.59513$
Root an. cond. $1.61094$
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  + (−1.15 − 1.99i)2-s + (0.5 + 0.866i)3-s + (−1.65 + 2.86i)4-s + (1.15 − 1.99i)6-s + (−0.5 + 0.866i)7-s + 2.99·8-s + (1 − 1.73i)9-s + (−0.802 − 1.39i)11-s − 3.30·12-s + 3.60·13-s + 2.30·14-s + (−0.151 − 0.262i)16-s + (3.80 − 6.58i)17-s − 4.60·18-s + (2.80 − 4.85i)19-s + ⋯
L(s)  = 1  + (−0.814 − 1.41i)2-s + (0.288 + 0.499i)3-s + (−0.825 + 1.43i)4-s + (0.470 − 0.814i)6-s + (−0.188 + 0.327i)7-s + 1.06·8-s + (0.333 − 0.577i)9-s + (−0.242 − 0.419i)11-s − 0.953·12-s + 1.00·13-s + 0.615·14-s + (−0.0378 − 0.0655i)16-s + (0.922 − 1.59i)17-s − 1.08·18-s + (0.643 − 1.11i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.252 + 0.967i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (276, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.252 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.565411 - 0.731987i\)
\(L(\frac12)\) \(\approx\) \(0.565411 - 0.731987i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 - 3.60T \)
good2 \( 1 + (1.15 + 1.99i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.802 + 1.39i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.80 + 6.58i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.80 + 4.85i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.10 - 5.37i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-1.80 - 3.12i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.10 - 8.84i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.21T + 47T^{2} \)
53 \( 1 - 3.21T + 53T^{2} \)
59 \( 1 + (-5.40 + 9.36i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.40 - 4.17i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.788T + 73T^{2} \)
79 \( 1 - 5.21T + 79T^{2} \)
83 \( 1 - 9.21T + 83T^{2} \)
89 \( 1 + (-3.10 - 5.37i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.19 - 7.26i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.26087904461792788765892937685, −10.35326202551668833516751277073, −9.481161302937695425632717230012, −9.037059217912272236157611903365, −8.034059161771232588227221268947, −6.59410422657347910222191309214, −5.00229982813411728945469653823, −3.50328552729800497329354268140, −2.85810061674438127217345055954, −0.977804594119646731272313460937, 1.52332187916338461010133311046, 3.76875283990016121837267682619, 5.44493730143086459700876841250, 6.26732699411462469006294834761, 7.33900165743966071498081908284, 7.972606871790070414945549898896, 8.597627009926907590430072397338, 9.961303153497257955958289818000, 10.39085008000391255936837556074, 11.96846106842072110437355093054

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