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Properties

Label 2-325-65.33-c1-0-4
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  + (0.880 − 1.52i)2-s + (−3.15 + 0.845i)3-s + (−0.550 − 0.953i)4-s + (−1.48 + 5.55i)6-s + (−1.20 + 0.694i)7-s + 1.58·8-s + (6.63 − 3.82i)9-s + (0.949 + 3.54i)11-s + (2.54 + 2.54i)12-s + (3.27 + 1.51i)13-s + 2.44i·14-s + (2.49 − 4.32i)16-s + (1.05 − 3.92i)17-s − 13.4i·18-s + (2.25 + 0.603i)19-s + ⋯
L(s)  = 1  + (0.622 − 1.07i)2-s + (−1.82 + 0.487i)3-s + (−0.275 − 0.476i)4-s + (−0.607 + 2.26i)6-s + (−0.454 + 0.262i)7-s + 0.559·8-s + (2.21 − 1.27i)9-s + (0.286 + 1.06i)11-s + (0.734 + 0.734i)12-s + (0.906 + 0.421i)13-s + 0.654i·14-s + (0.623 − 1.08i)16-s + (0.255 − 0.952i)17-s − 3.17i·18-s + (0.516 + 0.138i)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: $\chi_{325} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ 0.943 + 0.331i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10254 - 0.188303i\)
\(L(\frac12)\) \(\approx\) \(1.10254 - 0.188303i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.27 - 1.51i)T \)
good2 \( 1 + (-0.880 + 1.52i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (3.15 - 0.845i)T + (2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.20 - 0.694i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.949 - 3.54i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.05 + 3.92i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.25 - 0.603i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.47 - 5.51i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-2.47 - 1.42i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.220 - 0.220i)T + 31iT^{2} \)
37 \( 1 + (-5.26 - 3.03i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (10.9 - 2.94i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-5.35 - 1.43i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + 5.06iT - 47T^{2} \)
53 \( 1 + (-0.586 - 0.586i)T + 53iT^{2} \)
59 \( 1 + (-0.878 + 3.27i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.03 + 5.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.660 - 1.14i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.41 + 5.29i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 2.15T + 73T^{2} \)
79 \( 1 + 3.44iT - 79T^{2} \)
83 \( 1 + 1.38iT - 83T^{2} \)
89 \( 1 + (-1.09 + 0.293i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-7.92 - 13.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.66264118035349435821446932060, −10.99209388025057118774945888021, −10.00129997883685101509200636412, −9.475360233610310019476446657555, −7.35197973615000044117468452645, −6.41494402667755055295714607309, −5.25886327358942105040494931465, −4.51875778558014640030140930221, −3.41066550652494031711978449040, −1.37767376035042671905700009928, 1.02152798004786832934607576305, 3.92907213928020755871016666582, 5.12162892861963708518622687408, 6.06573523406375116910069971279, 6.32704877613300434455197140869, 7.33222540907480122464309693362, 8.418323499422158417869042657946, 10.28112588832064079167200819763, 10.80225443974835503980699543660, 11.69197577764117696150166050772

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