Properties

Degree $2$
Conductor $930$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 4·11-s + 12-s + 6·13-s + 15-s + 16-s + 2·17-s + 18-s + 4·19-s + 20-s − 4·22-s − 8·23-s + 24-s + 25-s + 6·26-s + 27-s + 6·29-s + 30-s − 31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s + 1.66·13-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s + 1.17·26-s + 0.192·27-s + 1.11·29-s + 0.182·30-s − 0.179·31-s + 0.176·32-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{930} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.243778481\)
\(L(\frac12)\) \(\approx\) \(3.243778481\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
31 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−19.93646369336296, −18.93221476051315, −18.18900302798427, −17.84442933304816, −16.50279019082772, −15.76039854826868, −15.67443141941896, −14.28849458543682, −14.00594854866973, −13.29106306918231, −12.66821573491162, −11.76377923887474, −10.79680150852365, −10.18699047326276, −9.276619296306586, −8.161080840551407, −7.717007596121910, −6.390415642610317, −5.765258289871271, −4.745186237846743, −3.611768193458522, −2.790331826256789, −1.525320635599925, 1.525320635599925, 2.790331826256789, 3.611768193458522, 4.745186237846743, 5.765258289871271, 6.390415642610317, 7.717007596121910, 8.161080840551407, 9.276619296306586, 10.18699047326276, 10.79680150852365, 11.76377923887474, 12.66821573491162, 13.29106306918231, 14.00594854866973, 14.28849458543682, 15.67443141941896, 15.76039854826868, 16.50279019082772, 17.84442933304816, 18.18900302798427, 18.93221476051315, 19.93646369336296

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