↑

Properties

Label 2-325-65.32-c1-0-8
Degree $2$
Conductor $325$
Sign $0.952 + 0.303i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.457 − 0.791i)2-s + (−0.0872 − 0.325i)3-s + (0.582 + 1.00i)4-s + (−0.297 − 0.0797i)6-s + (−3.61 + 2.08i)7-s + 2.89·8-s + (2.49 − 1.44i)9-s + (5.01 − 1.34i)11-s + (0.277 − 0.277i)12-s + (2.87 − 2.17i)13-s + 3.81i·14-s + (0.157 − 0.272i)16-s + (2.91 + 0.780i)17-s − 2.63i·18-s + (−1.43 + 5.35i)19-s + ⋯
L(s)  = 1  + (0.323 − 0.559i)2-s + (−0.0503 − 0.188i)3-s + (0.291 + 0.504i)4-s + (−0.121 − 0.0325i)6-s + (−1.36 + 0.788i)7-s + 1.02·8-s + (0.833 − 0.481i)9-s + (1.51 − 0.404i)11-s + (0.0801 − 0.0801i)12-s + (0.798 − 0.602i)13-s + 1.01i·14-s + (0.0393 − 0.0682i)16-s + (0.706 + 0.189i)17-s − 0.621i·18-s + (−0.329 + 1.22i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.952 + 0.303i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.952 + 0.303i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69145 - 0.263142i\)
\(L(\frac12)\) \(\approx\) \(1.69145 - 0.263142i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.87 + 2.17i)T \)
good2 \( 1 + (-0.457 + 0.791i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.0872 + 0.325i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (3.61 - 2.08i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.01 + 1.34i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.91 - 0.780i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (1.43 - 5.35i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.65 - 0.442i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (7.08 + 4.08i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.64 - 2.64i)T - 31iT^{2} \)
37 \( 1 + (6.72 + 3.88i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.228 + 0.853i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.224 + 0.839i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 - 1.44iT - 47T^{2} \)
53 \( 1 + (-0.405 + 0.405i)T - 53iT^{2} \)
59 \( 1 + (9.44 + 2.53i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.28 + 3.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.55 + 6.15i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.65 + 0.712i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + 6.02T + 73T^{2} \)
79 \( 1 + 12.5iT - 79T^{2} \)
83 \( 1 - 8.44iT - 83T^{2} \)
89 \( 1 + (-3.79 - 14.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.386 - 0.670i)T + (-48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.94234890544862758304284019336, −10.74817308192607233550801191920, −9.760932315282577762390478906970, −8.907633703439033043279297405119, −7.69478304863608945610528646874, −6.54156360554687096373693210175, −5.88655160221446406795935962482, −3.75860794825840082814629251288, −3.49464375803963405754560640912, −1.67810449411475874036893923948, 1.49819317429403450658870887881, 3.68178696660883574628483637280, 4.54292550495736211492113192082, 5.97549681241156764729094687538, 6.89149744559670241270418395818, 7.23545235484491982018631352625, 9.085918343338338788401567239431, 9.819494544158232611438477085764, 10.61438980864776782182094926627, 11.54362894267704836994166101954

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