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Properties

Label 2-325-65.4-c1-0-19
Degree $2$
Conductor $325$
Sign $-0.454 - 0.890i$
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  + (1.24 − 2.16i)2-s + (−2.44 − 1.41i)3-s + (−2.11 − 3.66i)4-s + (−6.10 + 3.52i)6-s + (−0.952 − 1.64i)7-s − 5.55·8-s + (2.49 + 4.32i)9-s + (0.926 + 0.534i)11-s + 11.9i·12-s + (3.32 − 1.40i)13-s − 4.75·14-s + (−2.70 + 4.69i)16-s + (−0.551 + 0.318i)17-s + 12.4·18-s + (−4.96 + 2.86i)19-s + ⋯
L(s)  = 1  + (0.882 − 1.52i)2-s + (−1.41 − 0.816i)3-s + (−1.05 − 1.83i)4-s + (−2.49 + 1.43i)6-s + (−0.360 − 0.623i)7-s − 1.96·8-s + (0.831 + 1.44i)9-s + (0.279 + 0.161i)11-s + 3.44i·12-s + (0.921 − 0.388i)13-s − 1.27·14-s + (−0.677 + 1.17i)16-s + (−0.133 + 0.0772i)17-s + 2.93·18-s + (−1.13 + 0.657i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.454 - 0.890i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.454 - 0.890i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.513449 + 0.838370i\)
\(L(\frac12)\) \(\approx\) \(0.513449 + 0.838370i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.32 + 1.40i)T \)
good2 \( 1 + (-1.24 + 2.16i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (2.44 + 1.41i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.952 + 1.64i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.926 - 0.534i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.551 - 0.318i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.96 - 2.86i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.30 + 1.90i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.72 + 8.18i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.46iT - 31T^{2} \)
37 \( 1 + (0.378 - 0.655i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.232 + 0.133i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.551 + 0.318i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 9.44T + 47T^{2} \)
53 \( 1 - 6.99iT - 53T^{2} \)
59 \( 1 + (-0.641 + 0.370i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.09 + 3.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.04 + 7.01i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.45 + 4.88i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 3.71T + 73T^{2} \)
79 \( 1 - 9.31T + 79T^{2} \)
83 \( 1 - 5.11T + 83T^{2} \)
89 \( 1 + (-10.8 - 6.28i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.11 + 3.65i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.10667762557894152041328814934, −10.60369724272773355968060793600, −9.795721942442599790654783454155, −8.082423670486066336247385778979, −6.50187500550325439291126130317, −5.97499575479133093528862978781, −4.70235439632025685941638708124, −3.74224231548805103257573395802, −1.97526660976212837833695099119, −0.64366625012979612894550011896, 3.63810305490161919611887494604, 4.62703860441439314705135119461, 5.43679630514590659838846593614, 6.33282342111358280773383379100, 6.73206235534048973156090323240, 8.371070430548554019093505101719, 9.213808346593715666889381766510, 10.53562258539878261706991946599, 11.47268546004352984444673767967, 12.35597671094200286074833469432

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