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Properties

Label 2-325-65.54-c2-0-0
Degree $2$
Conductor $325$
Sign $0.824 - 0.565i$
Analytic cond. $8.85560$
Root an. cond. $2.97583$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.01 − 3.77i)2-s + (−0.585 + 0.338i)3-s + (−9.76 − 5.63i)4-s + (0.684 + 2.55i)6-s + (1.21 + 4.53i)7-s + (−20.1 + 20.1i)8-s + (−4.27 + 7.39i)9-s + (1.84 − 6.88i)11-s + 7.63·12-s + (−12.9 + 0.564i)13-s + 18.3·14-s + (33.0 + 57.2i)16-s + (−9.76 + 16.9i)17-s + (23.6 + 23.6i)18-s + (2.43 + 9.10i)19-s + ⋯
L(s)  = 1  + (0.505 − 1.88i)2-s + (−0.195 + 0.112i)3-s + (−2.44 − 1.40i)4-s + (0.114 + 0.425i)6-s + (0.173 + 0.647i)7-s + (−2.51 + 2.51i)8-s + (−0.474 + 0.821i)9-s + (0.167 − 0.625i)11-s + 0.635·12-s + (−0.999 + 0.0434i)13-s + 1.31·14-s + (2.06 + 3.57i)16-s + (−0.574 + 0.994i)17-s + (1.31 + 1.31i)18-s + (0.128 + 0.478i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.824 - 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $0.824 - 0.565i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ 0.824 - 0.565i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.268601 + 0.0832671i\)
\(L(\frac12)\) \(\approx\) \(0.268601 + 0.0832671i\)
\(L(2)\) 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 \( 1 + (12.9 - 0.564i)T \)
good2 \( 1 + (-1.01 + 3.77i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (0.585 - 0.338i)T + (4.5 - 7.79i)T^{2} \)
7 \( 1 + (-1.21 - 4.53i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-1.84 + 6.88i)T + (-104. - 60.5i)T^{2} \)
17 \( 1 + (9.76 - 16.9i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-2.43 - 9.10i)T + (-312. + 180.5i)T^{2} \)
23 \( 1 + (12.1 + 21.0i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-0.898 - 1.55i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-14.7 + 14.7i)T - 961iT^{2} \)
37 \( 1 + (57.0 + 15.2i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (31.5 + 8.44i)T + (1.45e3 + 840.5i)T^{2} \)
43 \( 1 + (1.78 - 3.08i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (41.9 - 41.9i)T - 2.20e3iT^{2} \)
53 \( 1 - 79.2iT - 2.80e3T^{2} \)
59 \( 1 + (42.0 - 11.2i)T + (3.01e3 - 1.74e3i)T^{2} \)
61 \( 1 + (-14.4 + 24.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-1.43 + 5.37i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (-31.4 - 117. i)T + (-4.36e3 + 2.52e3i)T^{2} \)
73 \( 1 + (22.0 - 22.0i)T - 5.32e3iT^{2} \)
79 \( 1 - 40.0T + 6.24e3T^{2} \)
83 \( 1 + (102. + 102. i)T + 6.88e3iT^{2} \)
89 \( 1 + (-16.6 + 62.1i)T + (-6.85e3 - 3.96e3i)T^{2} \)
97 \( 1 + (33.3 - 8.92i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.54753431709504045615484999116, −10.68094222567914589038828962445, −10.09610426012929159441075628906, −8.922395102976894295514565678021, −8.238489039215163818830951745071, −6.01120503153922633149743664618, −5.14734028059471037325886732616, −4.17446105625026437496820816513, −2.82359884548956394528349685702, −1.87755919681963080915543206485, 0.11209150320318364341623166483, 3.41821714805497743589977208826, 4.65165718493420206309245678183, 5.36902659656449375389732855095, 6.75615981255466651826878386710, 7.03719631065362051279678697360, 8.118854107929678311469076559011, 9.163571057059933478095075433506, 9.905342275655087762689057941717, 11.73825320645029140326498959747

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