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Properties

Label 2-930-31.4-c1-0-6
Degree $2$
Conductor $930$
Sign $-0.249 - 0.968i$
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 + (0.606 − 0.440i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.309 − 0.951i)10-s + (−1.10 + 0.804i)11-s + (0.309 + 0.951i)12-s + (−1.77 + 5.46i)13-s + (0.606 + 0.440i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (1.27 + 0.926i)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 + (0.229 − 0.166i)7-s + (−0.286 − 0.207i)8-s + (−0.269 − 0.195i)9-s + (−0.0977 − 0.300i)10-s + (−0.333 + 0.242i)11-s + (0.0892 + 0.274i)12-s + (−0.492 + 1.51i)13-s + (0.162 + 0.117i)14-s + (−0.0797 + 0.245i)15-s + (0.0772 − 0.237i)16-s + (0.309 + 0.224i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.249 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.249 - 0.968i$
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: \((2,\ 930,\ (\ :1/2),\ -0.249 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.825483 + 1.06485i\)
\(L(\frac12)\) \(\approx\) \(0.825483 + 1.06485i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.16 - 4.58i)T \)
good7 \( 1 + (-0.606 + 0.440i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (1.10 - 0.804i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.77 - 5.46i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.27 - 0.926i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.19 - 3.68i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.89 - 2.10i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.76 - 5.42i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 4.40T + 37T^{2} \)
41 \( 1 + (2.68 + 8.25i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-1.62 - 5.01i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-0.985 + 3.03i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.64 + 1.92i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.952 - 2.93i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + 5.28T + 61T^{2} \)
67 \( 1 + 8.34T + 67T^{2} \)
71 \( 1 + (-0.633 - 0.460i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (9.58 - 6.96i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.40 - 3.19i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.24 + 3.83i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (0.422 - 0.307i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-11.3 + 8.24i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.21493429969376001707280695255, −9.212043484737808767392720127289, −8.491205370052677312046324117420, −7.54695005219025055005406311537, −7.10007749102541142262837460277, −6.17026722879495415687888396485, −5.05650787625571030893279333667, −4.20564698464815689886327378918, −3.05273870878946778924539842559, −1.53324783842402902794830311220, 0.60454462381447890905405069748, 2.60150753995296117633456049836, 3.21496144414825445098664958167, 4.49177427736364026307017416838, 5.12406131752696403357668855605, 6.12749905302268880793681393352, 7.60030395904902882723671624077, 8.187160086889175829046726163506, 9.159328886717558772314054176288, 9.991441919076609693512875039300

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