Properties

Label 2-930-93.92-c1-0-31
Degree $2$
Conductor $930$
Sign $-0.0522 + 0.998i$
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 + (−1.02 − 1.39i)3-s − 4-s + i·5-s + (1.39 − 1.02i)6-s + 0.608·7-s i·8-s + (−0.901 + 2.86i)9-s − 10-s − 1.47·11-s + (1.02 + 1.39i)12-s − 4.49i·13-s + 0.608i·14-s + (1.39 − 1.02i)15-s + 16-s − 0.826·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.591 − 0.806i)3-s − 0.5·4-s + 0.447i·5-s + (0.570 − 0.418i)6-s + 0.230·7-s − 0.353i·8-s + (−0.300 + 0.953i)9-s − 0.316·10-s − 0.444·11-s + (0.295 + 0.403i)12-s − 1.24i·13-s + 0.162i·14-s + (0.360 − 0.264i)15-s + 0.250·16-s − 0.200·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0522 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0522 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.0522 + 0.998i$
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: \((2,\ 930,\ (\ :1/2),\ -0.0522 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.434427 - 0.457745i\)
\(L(\frac12)\) \(\approx\) \(0.434427 - 0.457745i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.02 + 1.39i)T \)
5 \( 1 - iT \)
31 \( 1 + (4.31 + 3.52i)T \)
good7 \( 1 - 0.608T + 7T^{2} \)
11 \( 1 + 1.47T + 11T^{2} \)
13 \( 1 + 4.49iT - 13T^{2} \)
17 \( 1 + 0.826T + 17T^{2} \)
19 \( 1 + 0.806T + 19T^{2} \)
23 \( 1 - 1.31T + 23T^{2} \)
29 \( 1 + 1.72T + 29T^{2} \)
37 \( 1 - 2.79iT - 37T^{2} \)
41 \( 1 + 7.02iT - 41T^{2} \)
43 \( 1 + 12.7iT - 43T^{2} \)
47 \( 1 + 4.11iT - 47T^{2} \)
53 \( 1 + 7.69T + 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 + 7.71iT - 61T^{2} \)
67 \( 1 - 4.39T + 67T^{2} \)
71 \( 1 + 4.02iT - 71T^{2} \)
73 \( 1 - 4.96iT - 73T^{2} \)
79 \( 1 + 4.39iT - 79T^{2} \)
83 \( 1 + 2.04T + 83T^{2} \)
89 \( 1 + 12.1T + 89T^{2} \)
97 \( 1 - 2.89T + 97T^{2} \)
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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

−9.956542234256588712061332050287, −8.693749358817117229938261424268, −7.903113183131691236694784030988, −7.31524770410699679489339003186, −6.46809747579847810360856901741, −5.59577994836904961490054912034, −4.97885753971770570114332876552, −3.46758901991819306447576968692, −2.09502753681610111135441342168, −0.32712134098305778662505573679, 1.47435880814629783597474408557, 2.98476526110957139371033425346, 4.21403278610353157882805341045, 4.73679826245669841147553306149, 5.70303430466604414999832381033, 6.72073548327289236248374946189, 8.001452023503680625247849866301, 9.024567541873100727386068297946, 9.467333682752620103893415337042, 10.35697846699817492512623910393

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