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Properties

Label 2-930-31.8-c1-0-12
Degree $2$
Conductor $930$
Sign $0.999 + 0.0418i$
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 + (1.10 + 0.799i)7-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.309 − 0.951i)10-s + (4.05 + 2.94i)11-s + (0.309 − 0.951i)12-s + (−1.32 − 4.06i)13-s + (1.10 − 0.799i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (1.63 − 1.18i)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 + (0.415 + 0.302i)7-s + (−0.286 + 0.207i)8-s + (−0.269 + 0.195i)9-s + (0.0977 − 0.300i)10-s + (1.22 + 0.887i)11-s + (0.0892 − 0.274i)12-s + (−0.366 − 1.12i)13-s + (0.294 − 0.213i)14-s + (0.0797 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (0.395 − 0.287i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.999 + 0.0418i$
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: \((2,\ 930,\ (\ :1/2),\ 0.999 + 0.0418i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11025 - 0.0442107i\)
\(L(\frac12)\) \(\approx\) \(2.11025 - 0.0442107i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.14 - 2.13i)T \)
good7 \( 1 + (-1.10 - 0.799i)T + (2.16 + 6.65i)T^{2} \)
11 \( 1 + (-4.05 - 2.94i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.32 + 4.06i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.63 + 1.18i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.93 - 5.94i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-1.46 + 1.06i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.44 - 4.44i)T + (-23.4 - 17.0i)T^{2} \)
37 \( 1 - 9.73T + 37T^{2} \)
41 \( 1 + (-2.80 + 8.63i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (2.19 - 6.76i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (-0.604 - 1.85i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.78 - 3.47i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.37 + 10.3i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 - 3.12T + 61T^{2} \)
67 \( 1 - 15.4T + 67T^{2} \)
71 \( 1 + (12.3 - 8.93i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.15 - 0.835i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.49 + 1.08i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.94 + 9.07i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (0.742 + 0.539i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (7.19 + 5.22i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.980374037518231080367204484788, −9.546028554927493703863332332681, −8.596166099932784311173171762700, −7.73055768965324455175581990997, −6.43666768355552822595848069947, −5.47133522868483166868949630501, −4.65994581515966536308004396031, −3.69863498808335453024712770692, −2.59998079520583979720304298131, −1.41369312619809634236202152437, 1.10375828857476827514419466234, 2.55791925589574127418251831258, 3.94868990619167631980555528241, 4.79041114216365540779990296703, 6.10474893543842255403474476239, 6.52687341556232242604209117172, 7.43107924323083038881679414393, 8.336827109601252347582354841244, 9.114197625717928515728172573327, 9.724511150352745709533819839751

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