Properties

Label 2-930-1.1-c1-0-17
Degree $2$
Conductor $930$
Sign $1$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
Motivic weight $1$
Arithmetic yes
Rational no
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 + 3.37·7-s + 8-s + 9-s + 10-s + 0.627·11-s + 12-s − 2·13-s + 3.37·14-s + 15-s + 16-s − 4.74·17-s + 18-s − 0.627·19-s + 20-s + 3.37·21-s + 0.627·22-s + 3.37·23-s + 24-s + 25-s − 2·26-s + 27-s + 3.37·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.27·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.189·11-s + 0.288·12-s − 0.554·13-s + 0.901·14-s + 0.258·15-s + 0.250·16-s − 1.15·17-s + 0.235·18-s − 0.144·19-s + 0.223·20-s + 0.735·21-s + 0.133·22-s + 0.703·23-s + 0.204·24-s + 0.200·25-s − 0.392·26-s + 0.192·27-s + 0.637·28-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: $1$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.480187251\)
\(L(\frac12)\) \(\approx\) \(3.480187251\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.37T + 7T^{2} \)
11 \( 1 - 0.627T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 4.74T + 17T^{2} \)
19 \( 1 + 0.627T + 19T^{2} \)
23 \( 1 - 3.37T + 23T^{2} \)
29 \( 1 + 8.74T + 29T^{2} \)
37 \( 1 + 0.744T + 37T^{2} \)
41 \( 1 - 0.744T + 41T^{2} \)
43 \( 1 - 0.627T + 43T^{2} \)
47 \( 1 + 6.74T + 47T^{2} \)
53 \( 1 - 1.37T + 53T^{2} \)
59 \( 1 - 2.74T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 - 3.37T + 71T^{2} \)
73 \( 1 + 8.11T + 73T^{2} \)
79 \( 1 + 4.62T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + 1.37T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−10.10122555632368442110937507504, −9.151686987727392263319328945180, −8.383538037439911042697584044446, −7.45183342421001718272130718469, −6.70709419306710233180753915970, −5.48190214137245705985418789142, −4.76158486467898142834850524931, −3.85389128770315371773155464894, −2.50064394170691424181515377433, −1.68007154596548643645809445793, 1.68007154596548643645809445793, 2.50064394170691424181515377433, 3.85389128770315371773155464894, 4.76158486467898142834850524931, 5.48190214137245705985418789142, 6.70709419306710233180753915970, 7.45183342421001718272130718469, 8.383538037439911042697584044446, 9.151686987727392263319328945180, 10.10122555632368442110937507504

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