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Properties

Label 2-325-13.4-c1-0-12
Degree $2$
Conductor $325$
Sign $0.943 - 0.331i$
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  + (1.18 + 0.682i)2-s + (−0.199 + 0.346i)3-s + (−0.0676 − 0.117i)4-s + (−0.472 + 0.273i)6-s + (3.15 − 1.82i)7-s − 2.91i·8-s + (1.42 + 2.45i)9-s + (2.83 + 1.63i)11-s + 0.0541·12-s + (−3.53 − 0.710i)13-s + 4.97·14-s + (1.85 − 3.21i)16-s + (1.08 + 1.88i)17-s + 3.87i·18-s + (−0.742 + 0.428i)19-s + ⋯
L(s)  = 1  + (0.836 + 0.482i)2-s + (−0.115 + 0.199i)3-s + (−0.0338 − 0.0586i)4-s + (−0.193 + 0.111i)6-s + (1.19 − 0.688i)7-s − 1.03i·8-s + (0.473 + 0.819i)9-s + (0.855 + 0.494i)11-s + 0.0156·12-s + (−0.980 − 0.196i)13-s + 1.32·14-s + (0.463 − 0.803i)16-s + (0.263 + 0.456i)17-s + 0.914i·18-s + (−0.170 + 0.0983i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.943 - 0.331i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.943 - 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03175 + 0.346825i\)
\(L(\frac12)\) \(\approx\) \(2.03175 + 0.346825i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.53 + 0.710i)T \)
good2 \( 1 + (-1.18 - 0.682i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (0.199 - 0.346i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (-3.15 + 1.82i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.83 - 1.63i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.08 - 1.88i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.742 - 0.428i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.382 + 0.662i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.53 + 2.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.41iT - 31T^{2} \)
37 \( 1 + (9.40 + 5.42i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.59 + 3.80i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.91 + 3.32i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.68iT - 47T^{2} \)
53 \( 1 - 3.04T + 53T^{2} \)
59 \( 1 + (11.5 - 6.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.51 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.02 + 2.32i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.38 + 5.41i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 7.68iT - 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 - 1.18iT - 83T^{2} \)
89 \( 1 + (-2.82 - 1.63i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.54 + 2.62i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.86907087993147348954243781761, −10.49759575759307627933200666787, −10.16982004430162393220097569519, −8.755961757612692992668014456284, −7.48784606960989594981689981593, −6.88036893211946598755023933432, −5.37245010227582373181356958714, −4.73017746777613960126902073960, −3.89771678533305865997929437468, −1.64812897182605930389380657885, 1.79644424440539169782637807488, 3.24484262192834729247304971038, 4.47661542816419051661391703171, 5.28326403148275568201134029436, 6.54078715752308375569337763499, 7.80607580598812515461328635942, 8.734809748902688952762413770752, 9.674501320616906206793197491255, 11.15619830787556903697626969822, 11.85247793256731706496488674250

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