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Properties

Label 2-325-13.10-c1-0-1
Degree $2$
Conductor $325$
Sign $-0.0308 - 0.999i$
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  + (−0.576 + 0.333i)2-s + (−1.24 − 2.15i)3-s + (−0.778 + 1.34i)4-s + (1.43 + 0.826i)6-s + (−0.509 − 0.293i)7-s − 2.36i·8-s + (−1.58 + 2.74i)9-s + (−2.58 + 1.49i)11-s + 3.86·12-s + (1.93 + 3.04i)13-s + 0.391·14-s + (−0.767 − 1.32i)16-s + (−3.28 + 5.69i)17-s − 2.10i·18-s + (5.19 + 3.00i)19-s + ⋯
L(s)  = 1  + (−0.407 + 0.235i)2-s + (−0.716 − 1.24i)3-s + (−0.389 + 0.673i)4-s + (0.584 + 0.337i)6-s + (−0.192 − 0.111i)7-s − 0.837i·8-s + (−0.527 + 0.913i)9-s + (−0.779 + 0.450i)11-s + 1.11·12-s + (0.537 + 0.843i)13-s + 0.104·14-s + (−0.191 − 0.332i)16-s + (−0.797 + 1.38i)17-s − 0.496i·18-s + (1.19 + 0.688i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0308 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0308 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.0308 - 0.999i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.0308 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.302335 + 0.311817i\)
\(L(\frac12)\) \(\approx\) \(0.302335 + 0.311817i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.93 - 3.04i)T \)
good2 \( 1 + (0.576 - 0.333i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.24 + 2.15i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.509 + 0.293i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.58 - 1.49i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (3.28 - 5.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.19 - 3.00i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.335 + 0.580i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.94 - 6.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.53iT - 31T^{2} \)
37 \( 1 + (6.70 - 3.87i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.629 - 0.363i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.503 + 0.871i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 3.15iT - 47T^{2} \)
53 \( 1 + 4.94T + 53T^{2} \)
59 \( 1 + (12.3 + 7.13i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.03 + 6.98i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.551 + 0.318i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.79 + 4.50i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 16.8iT - 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 + 0.370iT - 83T^{2} \)
89 \( 1 + (8.00 - 4.61i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.99 - 3.46i)T + (48.5 + 84.0i)T^{2} \)
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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

−12.16728251113878850472227254381, −11.03581905136297673055756727653, −9.979427392940493374749987884799, −8.730075447544284483567330732125, −7.968848232847945169770616057115, −6.99578104794058895281144798998, −6.42161508115673811699147600293, −5.01080118074794965575641600363, −3.52407397370416959485843870329, −1.59308301723937351709846306801, 0.39948035839977568899718547009, 2.90293618392060249105318003144, 4.52309710822203375383839971422, 5.29440870590069453624130864434, 6.05252566176858776899169671964, 7.74828514669658074554632852774, 9.036904048898865830485699792289, 9.620455321099360241162484703176, 10.47493423834495962311310002523, 11.06062076742279988823700031637

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