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Properties

Label 2-1925-385.104-c0-0-3
Degree $2$
Conductor $1925$
Sign $0.499 + 0.866i$
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  + (0.198 − 0.0646i)2-s + (−0.773 + 0.562i)4-s + (−0.587 − 0.809i)7-s + (−0.240 + 0.330i)8-s + (−0.309 − 0.951i)9-s + (0.669 + 0.743i)11-s + (−0.169 − 0.122i)14-s + (0.269 − 0.828i)16-s + (−0.122 − 0.169i)18-s + (0.181 + 0.104i)22-s − 1.82i·23-s + (0.909 + 0.295i)28-s + (1.08 − 0.786i)29-s − 0.591i·32-s + (0.773 + 0.562i)36-s + (0.122 + 0.169i)37-s + ⋯
L(s)  = 1  + (0.198 − 0.0646i)2-s + (−0.773 + 0.562i)4-s + (−0.587 − 0.809i)7-s + (−0.240 + 0.330i)8-s + (−0.309 − 0.951i)9-s + (0.669 + 0.743i)11-s + (−0.169 − 0.122i)14-s + (0.269 − 0.828i)16-s + (−0.122 − 0.169i)18-s + (0.181 + 0.104i)22-s − 1.82i·23-s + (0.909 + 0.295i)28-s + (1.08 − 0.786i)29-s − 0.591i·32-s + (0.773 + 0.562i)36-s + (0.122 + 0.169i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.499 + 0.866i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (874, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ 0.499 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8792844607\)
\(L(\frac12)\) \(\approx\) \(0.8792844607\)
\(L(1)\) 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 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 + (-0.669 - 0.743i)T \)
good2 \( 1 + (-0.198 + 0.0646i)T + (0.809 - 0.587i)T^{2} \)
3 \( 1 + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + 1.82iT - T^{2} \)
29 \( 1 + (-1.08 + 0.786i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.122 - 0.169i)T + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + 1.33iT - T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + 1.95iT - T^{2} \)
71 \( 1 + (-0.413 + 1.27i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (-0.604 - 1.86i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.809 - 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.809 + 0.587i)T^{2} \)
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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

−9.245708538464342015500863164242, −8.589945510889122568456453547020, −7.74319281504669500854283618916, −6.74342719731948375050232462587, −6.31487932982232760291827582631, −4.96899314254969428091012642950, −4.17222988241712658431952976575, −3.61761555929739279722903931553, −2.56494106290740398511844325094, −0.67557532892651238843157288527, 1.39556226581192290970570505601, 2.81717550655938838123495958033, 3.71964849759041430748556537511, 4.79990855637797752916535910396, 5.58794035658278504016411438086, 6.04625680265722287079801895403, 7.06327385231331898663213444997, 8.215520911855402830929187212531, 8.774412438462872102548156985823, 9.472117122158326907853625062363

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