Properties

Label 2-930-1.1-c1-0-12
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 − 2.37·7-s + 8-s + 9-s + 10-s + 6.37·11-s + 12-s − 2·13-s − 2.37·14-s + 15-s + 16-s + 6.74·17-s + 18-s − 6.37·19-s + 20-s − 2.37·21-s + 6.37·22-s − 2.37·23-s + 24-s + 25-s − 2·26-s + 27-s − 2.37·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.896·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 1.92·11-s + 0.288·12-s − 0.554·13-s − 0.634·14-s + 0.258·15-s + 0.250·16-s + 1.63·17-s + 0.235·18-s − 1.46·19-s + 0.223·20-s − 0.517·21-s + 1.35·22-s − 0.494·23-s + 0.204·24-s + 0.200·25-s − 0.392·26-s + 0.192·27-s − 0.448·28-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

\(L(1)\) \(\approx\) \(3.178869393\)
\(L(\frac12)\) \(\approx\) \(3.178869393\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.37T + 7T^{2} \)
11 \( 1 - 6.37T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 6.74T + 17T^{2} \)
19 \( 1 + 6.37T + 19T^{2} \)
23 \( 1 + 2.37T + 23T^{2} \)
29 \( 1 - 2.74T + 29T^{2} \)
37 \( 1 - 10.7T + 37T^{2} \)
41 \( 1 + 10.7T + 41T^{2} \)
43 \( 1 - 6.37T + 43T^{2} \)
47 \( 1 - 4.74T + 47T^{2} \)
53 \( 1 + 4.37T + 53T^{2} \)
59 \( 1 + 8.74T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 + 0.744T + 67T^{2} \)
71 \( 1 + 2.37T + 71T^{2} \)
73 \( 1 - 9.11T + 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 - 4.37T + 89T^{2} \)
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

−9.845877171217146095031777213704, −9.456302010721688682340875744034, −8.432811589767721535782102862692, −7.35335038309971222476904616927, −6.43135534110010203914528061429, −5.96903423339873740311733271192, −4.52857870310880117502749956825, −3.71974408397570493107409839858, −2.79987541782877139287620011463, −1.49686357749362007261221879632, 1.49686357749362007261221879632, 2.79987541782877139287620011463, 3.71974408397570493107409839858, 4.52857870310880117502749956825, 5.96903423339873740311733271192, 6.43135534110010203914528061429, 7.35335038309971222476904616927, 8.432811589767721535782102862692, 9.456302010721688682340875744034, 9.845877171217146095031777213704

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