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Properties

Label 2-325-65.28-c1-0-16
Degree $2$
Conductor $325$
Sign $-0.154 + 0.987i$
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  + (1.58 − 0.915i)2-s + (0.512 − 1.91i)3-s + (0.677 − 1.17i)4-s + (−0.939 − 3.50i)6-s + (1.76 − 3.06i)7-s + 1.18i·8-s + (−0.803 − 0.463i)9-s + (−1.00 + 3.74i)11-s + (−1.89 − 1.89i)12-s + (−3.55 − 0.573i)13-s − 6.48i·14-s + (2.43 + 4.22i)16-s + (−1.95 + 0.524i)17-s − 1.69·18-s + (0.518 − 0.139i)19-s + ⋯
L(s)  = 1  + (1.12 − 0.647i)2-s + (0.296 − 1.10i)3-s + (0.338 − 0.586i)4-s + (−0.383 − 1.43i)6-s + (0.668 − 1.15i)7-s + 0.417i·8-s + (−0.267 − 0.154i)9-s + (−0.302 + 1.12i)11-s + (−0.548 − 0.548i)12-s + (−0.987 − 0.158i)13-s − 1.73i·14-s + (0.609 + 1.05i)16-s + (−0.474 + 0.127i)17-s − 0.400·18-s + (0.119 − 0.0319i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.154 + 0.987i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.154 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62849 - 1.90356i\)
\(L(\frac12)\) \(\approx\) \(1.62849 - 1.90356i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.55 + 0.573i)T \)
good2 \( 1 + (-1.58 + 0.915i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.512 + 1.91i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (-1.76 + 3.06i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.00 - 3.74i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.95 - 0.524i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.518 + 0.139i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.294 + 0.0788i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.71 - 0.988i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.13 + 4.13i)T - 31iT^{2} \)
37 \( 1 + (-2.70 - 4.69i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.649 - 0.174i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-2.28 - 8.51i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + 9.75T + 47T^{2} \)
53 \( 1 + (3.16 + 3.16i)T + 53iT^{2} \)
59 \( 1 + (-3.14 - 11.7i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.44 + 2.49i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.98 + 1.14i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.19 + 4.46i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + 14.7iT - 73T^{2} \)
79 \( 1 + 1.59iT - 79T^{2} \)
83 \( 1 - 7.57T + 83T^{2} \)
89 \( 1 + (-4.54 - 1.21i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (15.4 + 8.91i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.70246521276888897447963049786, −10.73045678254246615793563172137, −9.780484830430131608725665426462, −8.043463879313832649954500859581, −7.56908702664223032268059575799, −6.54013178897267320567367284352, −4.90648873261105980086495649282, −4.30361000051328745967395921142, −2.68680293959679542300263271047, −1.63119369525939013707534923875, 2.73923262926120511759620558689, 3.94666546397870435878157106666, 5.00098350579482093863203836060, 5.51673401032460269629149625043, 6.74471342815847932179500778012, 8.116159039015052029017172828316, 9.071210779783880274734127852876, 9.901401671580309928758185626727, 11.04206750131647595967663700213, 12.00402278725670996080891896737

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