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Properties

Label 2-1925-385.83-c0-0-6
Degree $2$
Conductor $1925$
Sign $0.978 + 0.208i$
Analytic cond. $0.960700$
Root an. cond. $0.980153$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.16 − 0.183i)3-s + (0.587 + 0.809i)4-s + (0.156 − 0.987i)7-s + (0.363 − 0.118i)9-s + (−0.309 − 0.951i)11-s + (0.831 + 0.831i)12-s + (0.734 − 1.44i)13-s + (−0.309 + 0.951i)16-s + (0.280 + 0.550i)17-s − 1.17i·21-s + (−0.647 + 0.329i)27-s + (0.891 − 0.453i)28-s + (−1.53 + 1.11i)29-s + (−0.533 − 1.04i)33-s + (0.309 + 0.224i)36-s + ⋯
L(s)  = 1  + (1.16 − 0.183i)3-s + (0.587 + 0.809i)4-s + (0.156 − 0.987i)7-s + (0.363 − 0.118i)9-s + (−0.309 − 0.951i)11-s + (0.831 + 0.831i)12-s + (0.734 − 1.44i)13-s + (−0.309 + 0.951i)16-s + (0.280 + 0.550i)17-s − 1.17i·21-s + (−0.647 + 0.329i)27-s + (0.891 − 0.453i)28-s + (−1.53 + 1.11i)29-s + (−0.533 − 1.04i)33-s + (0.309 + 0.224i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.978 + 0.208i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (468, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ 0.978 + 0.208i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.897202851\)
\(L(\frac12)\) \(\approx\) \(1.897202851\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + (-0.156 + 0.987i)T \)
11 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (-0.587 - 0.809i)T^{2} \)
3 \( 1 + (-1.16 + 0.183i)T + (0.951 - 0.309i)T^{2} \)
13 \( 1 + (-0.734 + 1.44i)T + (-0.587 - 0.809i)T^{2} \)
17 \( 1 + (-0.280 - 0.550i)T + (-0.587 + 0.809i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.951 - 0.309i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.297 - 1.87i)T + (-0.951 + 0.309i)T^{2} \)
53 \( 1 + (-0.587 - 0.809i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.809 - 0.587i)T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.610 + 0.0966i)T + (0.951 + 0.309i)T^{2} \)
79 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-1.44 + 0.734i)T + (0.587 - 0.809i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.863 + 1.69i)T + (-0.587 - 0.809i)T^{2} \)
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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

−9.017443715861773739854758864622, −8.428076123159130869218717001745, −7.74276262730874339510677640702, −7.49737671001327636076352551544, −6.31354374691960670681547488820, −5.44717324235561878876024836055, −3.92384761683630441541232026528, −3.39376568543135903939655598336, −2.76491654558222176374735511132, −1.43005586916215006319373486114, 1.93136434957506522467911336965, 2.21692622483491222058844665062, 3.41553740409049682800941933697, 4.48488835424899308263596296855, 5.45227399113869253560332919903, 6.23889608203218817186791074530, 7.14068651283221657079679632624, 7.892348425511757578957860005908, 8.873052170088508302341441634452, 9.351987838397067079096636722616

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