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Properties

Label 2-930-93.92-c1-0-16
Degree $2$
Conductor $930$
Sign $-0.136 - 0.990i$
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 + (−0.647 + 1.60i)3-s − 4-s − i·5-s + (−1.60 − 0.647i)6-s + 0.794·7-s − i·8-s + (−2.16 − 2.08i)9-s + 10-s + 6.29·11-s + (0.647 − 1.60i)12-s + 1.29i·13-s + 0.794i·14-s + (1.60 + 0.647i)15-s + 16-s + 3.58·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.374 + 0.927i)3-s − 0.5·4-s − 0.447i·5-s + (−0.655 − 0.264i)6-s + 0.300·7-s − 0.353i·8-s + (−0.720 − 0.693i)9-s + 0.316·10-s + 1.89·11-s + (0.187 − 0.463i)12-s + 0.359i·13-s + 0.212i·14-s + (0.414 + 0.167i)15-s + 0.250·16-s + 0.868·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.136 - 0.990i$
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: \((2,\ 930,\ (\ :1/2),\ -0.136 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.965110 + 1.10735i\)
\(L(\frac12)\) \(\approx\) \(0.965110 + 1.10735i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.647 - 1.60i)T \)
5 \( 1 + iT \)
31 \( 1 + (-4.83 + 2.76i)T \)
good7 \( 1 - 0.794T + 7T^{2} \)
11 \( 1 - 6.29T + 11T^{2} \)
13 \( 1 - 1.29iT - 13T^{2} \)
17 \( 1 - 3.58T + 17T^{2} \)
19 \( 1 + 1.52T + 19T^{2} \)
23 \( 1 + 3.08T + 23T^{2} \)
29 \( 1 - 7.82T + 29T^{2} \)
37 \( 1 - 6.43iT - 37T^{2} \)
41 \( 1 - 8.16iT - 41T^{2} \)
43 \( 1 - 5.37iT - 43T^{2} \)
47 \( 1 - 4.73iT - 47T^{2} \)
53 \( 1 + 3.60T + 53T^{2} \)
59 \( 1 + 12.3iT - 59T^{2} \)
61 \( 1 - 2.22iT - 61T^{2} \)
67 \( 1 + 2.25T + 67T^{2} \)
71 \( 1 + 15.0iT - 71T^{2} \)
73 \( 1 - 1.38iT - 73T^{2} \)
79 \( 1 - 11.5iT - 79T^{2} \)
83 \( 1 - 9.77T + 83T^{2} \)
89 \( 1 + 4.07T + 89T^{2} \)
97 \( 1 + 2.73T + 97T^{2} \)
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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

−9.868326257429469079962826954676, −9.583619329456089411319258988937, −8.612203392679373777348788541393, −7.984106704977941363904803498269, −6.41515289993662110686786006915, −6.25479310287076892176467300809, −4.84153780262998373635397634588, −4.38846937634762084503376412704, −3.35267129063050705312179083934, −1.18718460710743534832417415938, 0.953972350693627478230796573150, 2.01749282975472225068194842905, 3.26845618565610170521416635311, 4.34024806561843653139886571842, 5.59902050530897838571178110891, 6.43511191423464058058026968932, 7.22173861661201417699620628818, 8.249965428883562354725180084940, 8.949046363200266065110219773141, 10.08783393595432152631694754360

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