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Properties

Label 2-325-13.3-c1-0-16
Degree $2$
Conductor $325$
Sign $-0.938 + 0.344i$
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  + (−0.593 − 1.02i)2-s + (−0.172 − 0.298i)3-s + (0.295 − 0.511i)4-s + (−0.204 + 0.354i)6-s + (−1.01 + 1.75i)7-s − 3.07·8-s + (1.44 − 2.49i)9-s + (−1.94 − 3.36i)11-s − 0.203·12-s + (−2.05 − 2.96i)13-s + 2.40·14-s + (1.23 + 2.14i)16-s + (2.72 − 4.71i)17-s − 3.42·18-s + (−2.94 + 5.09i)19-s + ⋯
L(s)  = 1  + (−0.419 − 0.727i)2-s + (−0.0996 − 0.172i)3-s + (0.147 − 0.255i)4-s + (−0.0836 + 0.144i)6-s + (−0.383 + 0.664i)7-s − 1.08·8-s + (0.480 − 0.831i)9-s + (−0.585 − 1.01i)11-s − 0.0588·12-s + (−0.570 − 0.821i)13-s + 0.644·14-s + (0.308 + 0.535i)16-s + (0.660 − 1.14i)17-s − 0.806·18-s + (−0.674 + 1.16i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.938 + 0.344i$
Analytic conductor: \(2.59513\)
Root analytic conductor: \(1.61094\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (276, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1/2),\ -0.938 + 0.344i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.139797 - 0.787917i\)
\(L(\frac12)\) \(\approx\) \(0.139797 - 0.787917i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.05 + 2.96i)T \)
good2 \( 1 + (0.593 + 1.02i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.172 + 0.298i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.01 - 1.75i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.94 + 3.36i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.72 + 4.71i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.94 - 5.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.172 - 0.298i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.18T + 31T^{2} \)
37 \( 1 + (2.72 + 4.71i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0902 + 0.156i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.669 + 1.15i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 - 2.42T + 53T^{2} \)
59 \( 1 + (-3.53 + 6.11i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.38 - 5.85i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.20 - 3.81i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.940 + 1.62i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.86T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 7.83T + 83T^{2} \)
89 \( 1 + (6.12 + 10.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.90 + 5.02i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.13125253071217087359210589603, −10.22841312532384257519612718526, −9.559778928558465527687769786940, −8.638633188447255604235719372981, −7.42317738954907304048402086545, −6.09058589998675947301051008107, −5.49830758537521500291996246552, −3.49488531013592124361709342929, −2.45051787942386466549826812519, −0.63602583674095764609159264387, 2.30937594473853540571085422387, 3.98660700017505402262907067238, 5.10606803793250474266653497038, 6.59226293023253186200192898365, 7.24558589594377020642182015065, 8.004756555433423316418692953375, 9.151310406114313894047222586220, 10.12096053005127964203942581223, 10.85147033916777319802940478834, 12.15544673249689612898603577103

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