Properties

Label 2-930-1.1-c3-0-59
Degree $2$
Conductor $930$
Sign $-1$
Analytic cond. $54.8717$
Root an. cond. $7.40754$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3·3-s + 4·4-s − 5·5-s + 6·6-s + 26·7-s + 8·8-s + 9·9-s − 10·10-s − 48·11-s + 12·12-s − 88·13-s + 52·14-s − 15·15-s + 16·16-s − 54·17-s + 18·18-s − 160·19-s − 20·20-s + 78·21-s − 96·22-s − 48·23-s + 24·24-s + 25·25-s − 176·26-s + 27·27-s + 104·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.40·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.31·11-s + 0.288·12-s − 1.87·13-s + 0.992·14-s − 0.258·15-s + 1/4·16-s − 0.770·17-s + 0.235·18-s − 1.93·19-s − 0.223·20-s + 0.810·21-s − 0.930·22-s − 0.435·23-s + 0.204·24-s + 1/5·25-s − 1.32·26-s + 0.192·27-s + 0.701·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-1$
Analytic conductor: \(54.8717\)
Root analytic conductor: \(7.40754\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{930} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 930,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p T \)
3 \( 1 - p T \)
5 \( 1 + p T \)
31 \( 1 - p T \)
good7 \( 1 - 26 T + p^{3} T^{2} \)
11 \( 1 + 48 T + p^{3} T^{2} \)
13 \( 1 + 88 T + p^{3} T^{2} \)
17 \( 1 + 54 T + p^{3} T^{2} \)
19 \( 1 + 160 T + p^{3} T^{2} \)
23 \( 1 + 48 T + p^{3} T^{2} \)
29 \( 1 + 120 T + p^{3} T^{2} \)
37 \( 1 + 304 T + p^{3} T^{2} \)
41 \( 1 - 162 T + p^{3} T^{2} \)
43 \( 1 - 272 T + p^{3} T^{2} \)
47 \( 1 - 96 T + p^{3} T^{2} \)
53 \( 1 - 582 T + p^{3} T^{2} \)
59 \( 1 - 30 T + p^{3} T^{2} \)
61 \( 1 + 478 T + p^{3} T^{2} \)
67 \( 1 + 334 T + p^{3} T^{2} \)
71 \( 1 - 1062 T + p^{3} T^{2} \)
73 \( 1 - 212 T + p^{3} T^{2} \)
79 \( 1 + 640 T + p^{3} T^{2} \)
83 \( 1 - 972 T + p^{3} T^{2} \)
89 \( 1 - 120 T + p^{3} T^{2} \)
97 \( 1 - 86 T + p^{3} T^{2} \)
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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

−9.119609791196702552558972593859, −8.144241714271538468374827859541, −7.68360432394601083667149064837, −6.86770734752692927442033610688, −5.43312311932102694928202949088, −4.71167036938340332771645976958, −4.07788388684637798625410832278, −2.49659195379817187233620172379, −2.08408530339461023258800385508, 0, 2.08408530339461023258800385508, 2.49659195379817187233620172379, 4.07788388684637798625410832278, 4.71167036938340332771645976958, 5.43312311932102694928202949088, 6.86770734752692927442033610688, 7.68360432394601083667149064837, 8.144241714271538468374827859541, 9.119609791196702552558972593859

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