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Properties

Label 2-1925-385.118-c0-0-3
Degree $2$
Conductor $1925$
Sign $0.458 - 0.888i$
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  + (1.69 + 0.863i)3-s + (−0.951 − 0.309i)4-s + (0.453 + 0.891i)7-s + (1.53 + 2.11i)9-s + (0.809 − 0.587i)11-s + (−1.34 − 1.34i)12-s + (−0.610 − 0.0966i)13-s + (0.809 + 0.587i)16-s + (−1.59 + 0.253i)17-s + 1.90i·21-s + (0.481 + 3.03i)27-s + (−0.156 − 0.987i)28-s + (0.363 − 1.11i)29-s + (1.87 − 0.297i)33-s + (−0.809 − 2.48i)36-s + ⋯
L(s)  = 1  + (1.69 + 0.863i)3-s + (−0.951 − 0.309i)4-s + (0.453 + 0.891i)7-s + (1.53 + 2.11i)9-s + (0.809 − 0.587i)11-s + (−1.34 − 1.34i)12-s + (−0.610 − 0.0966i)13-s + (0.809 + 0.587i)16-s + (−1.59 + 0.253i)17-s + 1.90i·21-s + (0.481 + 3.03i)27-s + (−0.156 − 0.987i)28-s + (0.363 − 1.11i)29-s + (1.87 − 0.297i)33-s + (−0.809 − 2.48i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.458 - 0.888i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ 0.458 - 0.888i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.726765923\)
\(L(\frac12)\) \(\approx\) \(1.726765923\)
\(L(1)\) 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 \( 1 + (-0.453 - 0.891i)T \)
11 \( 1 + (-0.809 + 0.587i)T \)
good2 \( 1 + (0.951 + 0.309i)T^{2} \)
3 \( 1 + (-1.69 - 0.863i)T + (0.587 + 0.809i)T^{2} \)
13 \( 1 + (0.610 + 0.0966i)T + (0.951 + 0.309i)T^{2} \)
17 \( 1 + (1.59 - 0.253i)T + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.533 + 1.04i)T + (-0.587 - 0.809i)T^{2} \)
53 \( 1 + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (1.44 - 0.734i)T + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.0966 - 0.610i)T + (-0.951 + 0.309i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-1.16 - 0.183i)T + (0.951 + 0.309i)T^{2} \)
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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

−9.284038070014452946619840857324, −8.766793911455366032652682275361, −8.444949860940138246687082268138, −7.57151632806727041026913346077, −6.27619035025670510536847519255, −5.15912528042225851308682241725, −4.43612472028035241255423873245, −3.84553349658915747972108955044, −2.74943056105393757547020155692, −1.87301729891547212903587154126, 1.23065646004920517066899476006, 2.30383866691606490266577963627, 3.38287152191668897231405366149, 4.21457324813841845241348196892, 4.73448197026351073818801746216, 6.53344919218494758779887934237, 7.19979885383641552472340390144, 7.69674152141715837908348478418, 8.564734256648500586728924559485, 9.061810664126773618702083547993

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