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Properties

Label 2-325-65.64-c3-0-38
Degree $2$
Conductor $325$
Sign $0.996 + 0.0853i$
Analytic cond. $19.1756$
Root an. cond. $4.37899$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.170·2-s + 8.43i·3-s − 7.97·4-s − 1.43i·6-s + 32.7·7-s + 2.71·8-s − 44.1·9-s − 56.1i·11-s − 67.2i·12-s + (−17.3 − 43.5i)13-s − 5.57·14-s + 63.3·16-s − 67.8i·17-s + 7.50·18-s − 56.3i·19-s + ⋯
L(s)  = 1  − 0.0601·2-s + 1.62i·3-s − 0.996·4-s − 0.0976i·6-s + 1.76·7-s + 0.120·8-s − 1.63·9-s − 1.53i·11-s − 1.61i·12-s + (−0.369 − 0.929i)13-s − 0.106·14-s + 0.989·16-s − 0.968i·17-s + 0.0983·18-s − 0.680i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0853i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.996 + 0.0853i$
Analytic conductor: \(19.1756\)
Root analytic conductor: \(4.37899\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :3/2),\ 0.996 + 0.0853i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.414174489\)
\(L(\frac12)\) \(\approx\) \(1.414174489\)
\(L(\frac{5}{2})\) 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 \( 1 + (17.3 + 43.5i)T \)
good2 \( 1 + 0.170T + 8T^{2} \)
3 \( 1 - 8.43iT - 27T^{2} \)
7 \( 1 - 32.7T + 343T^{2} \)
11 \( 1 + 56.1iT - 1.33e3T^{2} \)
17 \( 1 + 67.8iT - 4.91e3T^{2} \)
19 \( 1 + 56.3iT - 6.85e3T^{2} \)
23 \( 1 - 31.9iT - 1.21e4T^{2} \)
29 \( 1 - 19.7T + 2.43e4T^{2} \)
31 \( 1 + 178. iT - 2.97e4T^{2} \)
37 \( 1 - 209.T + 5.06e4T^{2} \)
41 \( 1 - 126. iT - 6.89e4T^{2} \)
43 \( 1 + 109. iT - 7.95e4T^{2} \)
47 \( 1 + 468.T + 1.03e5T^{2} \)
53 \( 1 + 435. iT - 1.48e5T^{2} \)
59 \( 1 - 391. iT - 2.05e5T^{2} \)
61 \( 1 + 92.9T + 2.26e5T^{2} \)
67 \( 1 - 188.T + 3.00e5T^{2} \)
71 \( 1 - 394. iT - 3.57e5T^{2} \)
73 \( 1 - 596.T + 3.89e5T^{2} \)
79 \( 1 + 657.T + 4.93e5T^{2} \)
83 \( 1 - 560.T + 5.71e5T^{2} \)
89 \( 1 + 351. iT - 7.04e5T^{2} \)
97 \( 1 + 814.T + 9.12e5T^{2} \)
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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

−11.15061766281892479044761973986, −10.18254536360895963795475114477, −9.318823110207655567029619040834, −8.485788615119613070912235344702, −7.87693707843141992490967698499, −5.57840306556966249419058665648, −5.04147847651589370032645414679, −4.23949510829890125343481232081, −3.05557567277538039150018633288, −0.60287573092453937449055728504, 1.35166667452475675259149872875, 1.99910674953034886523875519861, 4.29473869353853731035423851873, 5.12192483026565623645843250886, 6.51060687945803677442685435796, 7.64016254787023198357668234084, 8.060268578179575138762289780657, 9.026574164891108513787022888091, 10.27449921459964055486403281449, 11.49682234931509382957620504767

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