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Properties

Label 2-930-31.4-c1-0-7
Degree $2$
Conductor $930$
Sign $0.978 + 0.207i$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)2-s + (0.309 − 0.951i)3-s + (−0.809 + 0.587i)4-s − 5-s − 0.999·6-s + (−3.70 + 2.69i)7-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)10-s + (4.06 − 2.95i)11-s + (0.309 + 0.951i)12-s + (−0.840 + 2.58i)13-s + (3.70 + 2.69i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (−2.30 − 1.67i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (0.178 − 0.549i)3-s + (−0.404 + 0.293i)4-s − 0.447·5-s − 0.408·6-s + (−1.40 + 1.01i)7-s + (0.286 + 0.207i)8-s + (−0.269 − 0.195i)9-s + (0.0977 + 0.300i)10-s + (1.22 − 0.891i)11-s + (0.0892 + 0.274i)12-s + (−0.233 + 0.717i)13-s + (0.989 + 0.719i)14-s + (−0.0797 + 0.245i)15-s + (0.0772 − 0.237i)16-s + (−0.560 − 0.406i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 + 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.978 + 0.207i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.978 + 0.207i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.03979 - 0.109152i\)
\(L(\frac12)\) \(\approx\) \(1.03979 - 0.109152i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + T \)
31 \( 1 + (-5.51 + 0.791i)T \)
good7 \( 1 + (3.70 - 2.69i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-4.06 + 2.95i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.840 - 2.58i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.30 + 1.67i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.382 - 1.17i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-7.52 - 5.46i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.119 + 0.368i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 + (-3.16 - 9.73i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-0.795 - 2.44i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (2.11 - 6.51i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.52 - 5.46i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.31 + 7.11i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 1.31T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 + (11.6 + 8.47i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.471 - 0.342i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (5.11 + 3.71i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.86 - 5.72i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (11.0 - 8.04i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (4.99 - 3.62i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.705044535377730736974443941121, −9.232669009025840263778306457916, −8.714638746913180262974741732091, −7.56455837215596420894384792243, −6.55225384291920772128863387482, −5.97832920493233643403796221592, −4.46948445624836775794035441911, −3.29493026245806084517961154149, −2.69536029065175775997768109512, −1.08887669300955302406309330287, 0.65539653024130172079224266086, 2.90891994019764467574893452404, 4.03645229796512780858648551711, 4.55278896514259900980837331948, 5.97914519421620268251402671434, 6.95257458987159003616470058845, 7.23112306740544197806334484028, 8.595714240984891780973624278513, 9.185062601221335505335045337653, 10.09056009239777739338611340165

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