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Properties

Label 2-1925-385.62-c0-0-6
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: $\chi_{1925} (832, \cdot )$
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

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

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