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Properties

Label 2-930-93.92-c1-0-18
Degree $2$
Conductor $930$
Sign $0.994 + 0.100i$
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  − i·2-s + (1.71 + 0.206i)3-s − 4-s + i·5-s + (0.206 − 1.71i)6-s − 1.18·7-s + i·8-s + (2.91 + 0.711i)9-s + 10-s − 1.57·11-s + (−1.71 − 0.206i)12-s + 3.43i·13-s + 1.18i·14-s + (−0.206 + 1.71i)15-s + 16-s + 4.55·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.992 + 0.119i)3-s − 0.5·4-s + 0.447i·5-s + (0.0844 − 0.702i)6-s − 0.446·7-s + 0.353i·8-s + (0.971 + 0.237i)9-s + 0.316·10-s − 0.475·11-s + (−0.496 − 0.0596i)12-s + 0.953i·13-s + 0.315i·14-s + (−0.0533 + 0.444i)15-s + 0.250·16-s + 1.10·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.994 + 0.100i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.994 + 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09569 - 0.105579i\)
\(L(\frac12)\) \(\approx\) \(2.09569 - 0.105579i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-1.71 - 0.206i)T \)
5 \( 1 - iT \)
31 \( 1 + (5.56 - 0.105i)T \)
good7 \( 1 + 1.18T + 7T^{2} \)
11 \( 1 + 1.57T + 11T^{2} \)
13 \( 1 - 3.43iT - 13T^{2} \)
17 \( 1 - 4.55T + 17T^{2} \)
19 \( 1 - 6.64T + 19T^{2} \)
23 \( 1 - 1.16T + 23T^{2} \)
29 \( 1 - 8.20T + 29T^{2} \)
37 \( 1 - 9.82iT - 37T^{2} \)
41 \( 1 + 5.42iT - 41T^{2} \)
43 \( 1 + 6.82iT - 43T^{2} \)
47 \( 1 - 1.46iT - 47T^{2} \)
53 \( 1 - 7.16T + 53T^{2} \)
59 \( 1 - 6.84iT - 59T^{2} \)
61 \( 1 - 11.8iT - 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 + 6.52iT - 71T^{2} \)
73 \( 1 + 4.67iT - 73T^{2} \)
79 \( 1 + 6.13iT - 79T^{2} \)
83 \( 1 - 3.86T + 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 - 3.46T + 97T^{2} \)
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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

−10.04998162374218119812397537968, −9.365246380720600544915352063033, −8.583675756373418500895296984126, −7.59913637019183096893321626625, −6.91018572057334117487010750428, −5.53753785237939754238886167488, −4.43020455438681106182905854625, −3.34081067333359446811334215393, −2.80273006729021991871760009389, −1.44852276090241037486706787858, 1.04398508478023221137766804694, 2.85707349447426191020522937357, 3.62871850861648337862229737068, 4.94148236387645159430872184753, 5.69949644719968593176899402464, 6.88624198704910494745961790948, 7.79267857474204427313188217096, 8.121552313155874631143094818849, 9.242222100765257340556378164590, 9.737328738295501416180693031702

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