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Properties

Label 2-325-13.11-c2-0-23
Degree $2$
Conductor $325$
Sign $-0.328 + 0.944i$
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  + (0.723 − 2.69i)2-s + (−0.0169 − 0.0293i)3-s + (−3.29 − 1.90i)4-s + (−0.0915 + 0.0245i)6-s + (2.46 + 9.18i)7-s + (0.377 − 0.377i)8-s + (4.49 − 7.79i)9-s + (13.6 + 3.66i)11-s + 0.129i·12-s + (4.73 − 12.1i)13-s + 26.5·14-s + (−8.36 − 14.4i)16-s + (−17.8 − 10.3i)17-s + (−17.7 − 17.7i)18-s + (13.8 − 3.70i)19-s + ⋯
L(s)  = 1  + (0.361 − 1.34i)2-s + (−0.00565 − 0.00979i)3-s + (−0.824 − 0.476i)4-s + (−0.0152 + 0.00408i)6-s + (0.351 + 1.31i)7-s + (0.0471 − 0.0471i)8-s + (0.499 − 0.865i)9-s + (1.24 + 0.332i)11-s + 0.0107i·12-s + (0.364 − 0.931i)13-s + 1.89·14-s + (−0.522 − 0.905i)16-s + (−1.05 − 0.606i)17-s + (−0.987 − 0.987i)18-s + (0.728 − 0.195i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(325\)    =    \(5^{2} \cdot 13\)
Sign: $-0.328 + 0.944i$
Analytic conductor: \(8.85560\)
Root analytic conductor: \(2.97583\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{325} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 325,\ (\ :1),\ -0.328 + 0.944i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.32417 - 1.86149i\)
\(L(\frac12)\) \(\approx\) \(1.32417 - 1.86149i\)
\(L(2)\) 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 \( 1 + (-4.73 + 12.1i)T \)
good2 \( 1 + (-0.723 + 2.69i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (0.0169 + 0.0293i)T + (-4.5 + 7.79i)T^{2} \)
7 \( 1 + (-2.46 - 9.18i)T + (-42.4 + 24.5i)T^{2} \)
11 \( 1 + (-13.6 - 3.66i)T + (104. + 60.5i)T^{2} \)
17 \( 1 + (17.8 + 10.3i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-13.8 + 3.70i)T + (312. - 180.5i)T^{2} \)
23 \( 1 + (16.7 - 9.66i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (6.94 + 12.0i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (-38.2 - 38.2i)T + 961iT^{2} \)
37 \( 1 + (-17.7 - 4.75i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + (-13.2 + 49.5i)T + (-1.45e3 - 840.5i)T^{2} \)
43 \( 1 + (-1.88 - 1.08i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-12.8 + 12.8i)T - 2.20e3iT^{2} \)
53 \( 1 + 31.1T + 2.80e3T^{2} \)
59 \( 1 + (-12.6 - 47.0i)T + (-3.01e3 + 1.74e3i)T^{2} \)
61 \( 1 + (-0.160 + 0.278i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (11.5 - 43.0i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + (71.6 - 19.1i)T + (4.36e3 - 2.52e3i)T^{2} \)
73 \( 1 + (2.91 - 2.91i)T - 5.32e3iT^{2} \)
79 \( 1 + 10.8T + 6.24e3T^{2} \)
83 \( 1 + (50.3 + 50.3i)T + 6.88e3iT^{2} \)
89 \( 1 + (-9.10 - 2.43i)T + (6.85e3 + 3.96e3i)T^{2} \)
97 \( 1 + (115. - 31.0i)T + (8.14e3 - 4.70e3i)T^{2} \)
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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

−11.54642637151621078796761714144, −10.28426271645183821237904144181, −9.404844126076309257101805022123, −8.744467272850264927729018744777, −7.17908403700118870818205512678, −6.01101860246151083375865559878, −4.71455481552251215404900063639, −3.63074241998668973485137700318, −2.47940167241601308679869528329, −1.17040230723129662090769664813, 1.59579084358973075076503595226, 4.16635855965969944948347163282, 4.47771241818441593303030426477, 6.08816817997695721359969404328, 6.78548856854780932217893661133, 7.64988653587205272398145348271, 8.440431653219759875354537381781, 9.679117912710080192976974505169, 10.87075446887905172869995057957, 11.48179449795583774644778125620

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