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Properties

Label 2-1925-385.167-c0-0-4
Degree $2$
Conductor $1925$
Sign $-0.998 - 0.0483i$
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  + (−0.829 − 1.62i)2-s + (−1.37 + 1.89i)4-s + (0.987 − 0.156i)7-s + (2.41 + 0.382i)8-s + (−0.951 − 0.309i)9-s + (−0.978 − 0.207i)11-s + (−1.07 − 1.47i)14-s + (−0.657 − 2.02i)16-s + (0.285 + 1.80i)18-s + (0.472 + 1.76i)22-s + (1.40 − 1.40i)23-s + (−1.06 + 2.08i)28-s + (−0.336 − 0.244i)29-s + (−1.02 + 1.02i)32-s + (1.89 − 1.37i)36-s + (−0.127 − 0.803i)37-s + ⋯
L(s)  = 1  + (−0.829 − 1.62i)2-s + (−1.37 + 1.89i)4-s + (0.987 − 0.156i)7-s + (2.41 + 0.382i)8-s + (−0.951 − 0.309i)9-s + (−0.978 − 0.207i)11-s + (−1.07 − 1.47i)14-s + (−0.657 − 2.02i)16-s + (0.285 + 1.80i)18-s + (0.472 + 1.76i)22-s + (1.40 − 1.40i)23-s + (−1.06 + 2.08i)28-s + (−0.336 − 0.244i)29-s + (−1.02 + 1.02i)32-s + (1.89 − 1.37i)36-s + (−0.127 − 0.803i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0483i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0483i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $-0.998 - 0.0483i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ -0.998 - 0.0483i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5426136580\)
\(L(\frac12)\) \(\approx\) \(0.5426136580\)
\(L(1)\) 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 \( 1 + (-0.987 + 0.156i)T \)
11 \( 1 + (0.978 + 0.207i)T \)
good2 \( 1 + (0.829 + 1.62i)T + (-0.587 + 0.809i)T^{2} \)
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 \( 1 + (-1.40 + 1.40i)T - iT^{2} \)
29 \( 1 + (0.336 + 0.244i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.127 + 0.803i)T + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (1.38 + 1.38i)T + iT^{2} \)
47 \( 1 + (-0.951 - 0.309i)T^{2} \)
53 \( 1 + (0.863 + 1.69i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-1.05 - 1.05i)T + iT^{2} \)
71 \( 1 + (0.604 + 1.86i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.459 - 1.41i)T + (-0.809 - 0.587i)T^{2} \)
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

−8.967375541376915439446360781318, −8.476191865759650268771808675080, −7.961525909202127824309128356205, −6.90547891143460823300730333784, −5.45379050336341204024850209487, −4.71690076765565368492029026279, −3.60849842612982162813775948046, −2.75808551388734063479715326801, −1.96044972545278946795175210203, −0.54884305167697252103869876261, 1.45450417016302923103022690260, 2.97080300180099711354587066767, 4.71737403717496743347163983604, 5.21465827223841500272231921919, 5.81324058803140017180187328058, 6.81841723780436568126742438178, 7.70065618008043362662696672113, 7.997881385278636840566778995253, 8.788700999171882020387772105300, 9.385656837369486600579159180191

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