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Properties

Label 2-1925-385.13-c0-0-7
Degree $2$
Conductor $1925$
Sign $0.403 + 0.915i$
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  + (1.62 − 0.829i)2-s + (1.37 − 1.89i)4-s + (0.156 + 0.987i)7-s + (0.382 − 2.41i)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 + (1.80 − 0.285i)18-s + (−1.76 + 0.472i)22-s + (−1.40 − 1.40i)23-s + (2.08 + 1.06i)28-s + (0.336 + 0.244i)29-s + (−1.02 − 1.02i)32-s + (1.89 − 1.37i)36-s + (0.803 − 0.127i)37-s + ⋯
L(s)  = 1  + (1.62 − 0.829i)2-s + (1.37 − 1.89i)4-s + (0.156 + 0.987i)7-s + (0.382 − 2.41i)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 + (1.80 − 0.285i)18-s + (−1.76 + 0.472i)22-s + (−1.40 − 1.40i)23-s + (2.08 + 1.06i)28-s + (0.336 + 0.244i)29-s + (−1.02 − 1.02i)32-s + (1.89 − 1.37i)36-s + (0.803 − 0.127i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.403 + 0.915i$
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: \((2,\ 1925,\ (\ :0),\ 0.403 + 0.915i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.904744627\)
\(L(\frac12)\) \(\approx\) \(2.904744627\)
\(L(1)\) 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 \( 1 + (0.978 + 0.207i)T \)
good2 \( 1 + (-1.62 + 0.829i)T + (0.587 - 0.809i)T^{2} \)
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 \( 1 + (-0.803 + 0.127i)T + (0.951 - 0.309i)T^{2} \)
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 \( 1 + (1.69 - 0.863i)T + (0.587 - 0.809i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (1.05 - 1.05i)T - iT^{2} \)
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

−9.621453481532464063038527841254, −8.401427256491278666625409665503, −7.58393210681997290767759454465, −6.32327604867990663026176320818, −5.94850034366375996925301667209, −4.74824250101709978507275650316, −4.62490223080984793842935441111, −3.29619739986209508092158653902, −2.50070561250077926963129384972, −1.69680520173977908526995956680, 1.87758443014861005029351416859, 3.24861573857621369576004002326, 3.96988227503554251981865102182, 4.64993823155729443176378546447, 5.41456288431604379588053822507, 6.31681147705155293189956953740, 7.05930268101541608446049774228, 7.65363868325684872399049190455, 8.181509994375276195776229639171, 9.727475573307881683476211058745

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