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Properties

Label 2-930-31.4-c1-0-11
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: $\chi_{930} (841, \cdot )$
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

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

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