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Properties

Label 2-930-31.8-c1-0-10
Degree $2$
Conductor $930$
Sign $0.957 - 0.286i$
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 + (0.545 + 0.396i)7-s + (0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.309 + 0.951i)10-s + (0.613 + 0.445i)11-s + (−0.309 + 0.951i)12-s + (0.456 + 1.40i)13-s + (−0.545 + 0.396i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (5.64 − 4.10i)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 + (0.206 + 0.149i)7-s + (0.286 − 0.207i)8-s + (−0.269 + 0.195i)9-s + (−0.0977 + 0.300i)10-s + (0.185 + 0.134i)11-s + (−0.0892 + 0.274i)12-s + (0.126 + 0.389i)13-s + (−0.145 + 0.105i)14-s + (−0.0797 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (1.36 − 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.957 - 0.286i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.957 - 0.286i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43752 + 0.210705i\)
\(L(\frac12)\) \(\approx\) \(1.43752 + 0.210705i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.55 - 0.338i)T \)
good7 \( 1 + (-0.545 - 0.396i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-0.613 - 0.445i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.456 - 1.40i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-5.64 + 4.10i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.439 + 1.35i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (5.79 - 4.21i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.10 - 3.39i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 - 6.68T + 37T^{2} \)
41 \( 1 + (-3.27 + 10.0i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-3.45 + 10.6i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-4.03 - 12.4i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.45 + 1.05i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.89 - 5.83i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 - 4.47T + 61T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + (-4.12 + 2.99i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-9.62 - 6.99i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.74 - 2.72i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.472 + 1.45i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (7.16 + 5.20i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.554 - 0.403i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.833829888252730914627021075001, −9.292108445866530382570571316895, −8.269102293993528621726133199141, −7.49285968585623705202773226207, −6.80124385255278117421365708326, −5.76832222918936948565439822487, −5.25176838651945780565808659557, −3.91811216601269506117533464088, −2.40176426559253492151520758974, −1.03857105223238681396082165081, 1.08270866553385085361584840355, 2.51857959197043948862167493642, 3.67434529708787201135282648475, 4.50566677305292386530455946595, 5.67410240153010042302119779363, 6.34616342309345844543269975012, 7.983492625945397177265923134931, 8.266297919684502928847842680752, 9.712747111108031761365065771380, 9.869757913037511910495919789251

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