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Properties

Label 2-1925-385.13-c0-0-5
Degree $2$
Conductor $1925$
Sign $0.994 + 0.108i$
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  + (0.186 − 0.0949i)2-s + (−0.562 + 0.773i)4-s + (−0.156 − 0.987i)7-s + (−0.0639 + 0.403i)8-s + (0.951 + 0.309i)9-s + (0.669 − 0.743i)11-s + (−0.122 − 0.169i)14-s + (−0.269 − 0.828i)16-s + (0.206 − 0.0327i)18-s + (0.0541 − 0.201i)22-s + (−0.575 − 0.575i)23-s + (0.852 + 0.434i)28-s + (1.20 + 0.873i)29-s + (−0.417 − 0.417i)32-s + (−0.773 + 0.562i)36-s + (1.96 − 0.311i)37-s + ⋯
L(s)  = 1  + (0.186 − 0.0949i)2-s + (−0.562 + 0.773i)4-s + (−0.156 − 0.987i)7-s + (−0.0639 + 0.403i)8-s + (0.951 + 0.309i)9-s + (0.669 − 0.743i)11-s + (−0.122 − 0.169i)14-s + (−0.269 − 0.828i)16-s + (0.206 − 0.0327i)18-s + (0.0541 − 0.201i)22-s + (−0.575 − 0.575i)23-s + (0.852 + 0.434i)28-s + (1.20 + 0.873i)29-s + (−0.417 − 0.417i)32-s + (−0.773 + 0.562i)36-s + (1.96 − 0.311i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.994 + 0.108i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1168, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ 0.994 + 0.108i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.209887579\)
\(L(\frac12)\) \(\approx\) \(1.209887579\)
\(L(1)\) 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 \( 1 + (-0.669 + 0.743i)T \)
good2 \( 1 + (-0.186 + 0.0949i)T + (0.587 - 0.809i)T^{2} \)
3 \( 1 + (-0.951 - 0.309i)T^{2} \)
13 \( 1 + (0.587 - 0.809i)T^{2} \)
17 \( 1 + (0.587 + 0.809i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.575 + 0.575i)T + iT^{2} \)
29 \( 1 + (-1.20 - 0.873i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-1.96 + 0.311i)T + (0.951 - 0.309i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (0.946 - 0.946i)T - iT^{2} \)
47 \( 1 + (0.951 + 0.309i)T^{2} \)
53 \( 1 + (-1.69 + 0.863i)T + (0.587 - 0.809i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.294 + 0.294i)T - iT^{2} \)
71 \( 1 + (-0.413 - 1.27i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.951 + 0.309i)T^{2} \)
79 \( 1 + (-0.128 + 0.395i)T + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (-0.587 - 0.809i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.587 - 0.809i)T^{2} \)
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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

−9.426947575043655650640020045032, −8.445513108987106263110077317140, −7.916935689608912787982867900301, −7.03319831697018757601129626126, −6.38993199763041907961645539929, −5.08659836328243264800190077706, −4.24413358003608574621568552188, −3.77985772540070400292340388078, −2.69553635178027633528427060879, −1.08846533663910260608461259317, 1.27190641335690060489544059500, 2.38192525104131651747149792956, 3.86363065471730022691240025195, 4.49603402824431292703607811425, 5.38654210499195930083753773205, 6.24204735497243606253820778993, 6.78082905234289995421680256092, 7.88825902666989608228646588909, 8.804014403936434352328792652678, 9.626412160647037295876602174796

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