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Properties

Label 2-325-65.37-c1-0-0
Degree $2$
Conductor $325$
Sign $-0.954 - 0.299i$
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  + (0.113 − 0.0656i)2-s + (−0.332 − 0.0890i)3-s + (−0.991 + 1.71i)4-s + (−0.0436 + 0.0116i)6-s + (−1.39 + 2.40i)7-s + 0.522i·8-s + (−2.49 − 1.44i)9-s + (−3.91 − 1.04i)11-s + (0.482 − 0.482i)12-s + (−0.756 − 3.52i)13-s + 0.365i·14-s + (−1.94 − 3.37i)16-s + (0.627 + 2.34i)17-s − 0.378·18-s + (0.491 + 1.83i)19-s + ⋯
L(s)  = 1  + (0.0804 − 0.0464i)2-s + (−0.191 − 0.0513i)3-s + (−0.495 + 0.858i)4-s + (−0.0178 + 0.00477i)6-s + (−0.525 + 0.910i)7-s + 0.184i·8-s + (−0.831 − 0.480i)9-s + (−1.18 − 0.316i)11-s + (0.139 − 0.139i)12-s + (−0.209 − 0.977i)13-s + 0.0976i·14-s + (−0.487 − 0.843i)16-s + (0.152 + 0.567i)17-s − 0.0891·18-s + (0.112 + 0.420i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.954 - 0.299i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (232, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.954 - 0.299i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0575670 + 0.376190i\)
\(L(\frac12)\) \(\approx\) \(0.0575670 + 0.376190i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.756 + 3.52i)T \)
good2 \( 1 + (-0.113 + 0.0656i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.332 + 0.0890i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (1.39 - 2.40i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.91 + 1.04i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.627 - 2.34i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.491 - 1.83i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.06 - 7.70i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (3.96 - 2.28i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.87 - 3.87i)T + 31iT^{2} \)
37 \( 1 + (3.50 + 6.07i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.66 - 6.20i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-6.24 + 1.67i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 - 0.512T + 47T^{2} \)
53 \( 1 + (-1.32 + 1.32i)T - 53iT^{2} \)
59 \( 1 + (-2.53 + 0.679i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.641 + 1.11i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.13 - 1.80i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.20 - 1.66i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 - 9.93iT - 73T^{2} \)
79 \( 1 + 8.37iT - 79T^{2} \)
83 \( 1 + 3.17T + 83T^{2} \)
89 \( 1 + (-1.61 + 6.01i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (10.1 + 5.88i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.21192605493600410069738329180, −11.29294184930257175649271893680, −10.12208238781081104460047253862, −9.079897622347669221253432898960, −8.299264892921809500254820153702, −7.47799942336078469483336576615, −5.85058510545848997698074446900, −5.32631494198363884964118670594, −3.54213228535832708064869995947, −2.76534344515475590296949975106, 0.25543617329791832722533802898, 2.46609576067509666542985565005, 4.26564393573311409990374076082, 5.10997458869616735971109741876, 6.20110532163396780269139030980, 7.22788280693068384721455939662, 8.422522960181157880826814682500, 9.519859842001835196774141919792, 10.31823472256450661993978087141, 10.92832584714929396424361090074

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