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Properties

Label 2-930-93.92-c1-0-19
Degree $2$
Conductor $930$
Sign $0.999 + 0.0213i$
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 + (−1.14 − 1.29i)3-s − 4-s − i·5-s + (1.29 − 1.14i)6-s + 3.69·7-s − i·8-s + (−0.365 + 2.97i)9-s + 10-s − 0.851·11-s + (1.14 + 1.29i)12-s + 2.29i·13-s + 3.69i·14-s + (−1.29 + 1.14i)15-s + 16-s − 1.10·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.662 − 0.748i)3-s − 0.5·4-s − 0.447i·5-s + (0.529 − 0.468i)6-s + 1.39·7-s − 0.353i·8-s + (−0.121 + 0.992i)9-s + 0.316·10-s − 0.256·11-s + (0.331 + 0.374i)12-s + 0.636i·13-s + 0.987i·14-s + (−0.334 + 0.296i)15-s + 0.250·16-s − 0.268·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.999 + 0.0213i$
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: \((2,\ 930,\ (\ :1/2),\ 0.999 + 0.0213i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37113 - 0.0146081i\)
\(L(\frac12)\) \(\approx\) \(1.37113 - 0.0146081i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.14 + 1.29i)T \)
5 \( 1 + iT \)
31 \( 1 + (-3.77 + 4.09i)T \)
good7 \( 1 - 3.69T + 7T^{2} \)
11 \( 1 + 0.851T + 11T^{2} \)
13 \( 1 - 2.29iT - 13T^{2} \)
17 \( 1 + 1.10T + 17T^{2} \)
19 \( 1 - 4.96T + 19T^{2} \)
23 \( 1 + 1.74T + 23T^{2} \)
29 \( 1 - 4.78T + 29T^{2} \)
37 \( 1 + 2.69iT - 37T^{2} \)
41 \( 1 + 1.95iT - 41T^{2} \)
43 \( 1 + 1.65iT - 43T^{2} \)
47 \( 1 + 4.65iT - 47T^{2} \)
53 \( 1 - 13.1T + 53T^{2} \)
59 \( 1 - 7.91iT - 59T^{2} \)
61 \( 1 - 3.10iT - 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 - 0.0953iT - 71T^{2} \)
73 \( 1 + 16.2iT - 73T^{2} \)
79 \( 1 - 13.5iT - 79T^{2} \)
83 \( 1 - 14.0T + 83T^{2} \)
89 \( 1 - 3.95T + 89T^{2} \)
97 \( 1 - 6.65T + 97T^{2} \)
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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

−10.07779158197497844013976312331, −8.911666180875242991123727622120, −8.150298915863748074480521423152, −7.55458961539174389229385380428, −6.72227708100989934973292640824, −5.65259909455050957827135140832, −5.04883535053497721289807352346, −4.21717922017397362603903482314, −2.19283511226903300283033015736, −0.957874007402229992505032918176, 1.08757407565924008503098333176, 2.66366932113884602831816216516, 3.76346895705452771555048787714, 4.84752030657233746096621293590, 5.31198952976812528621337190222, 6.49462528782645986976807380310, 7.74898049348778483128820482530, 8.483747173535752789686515394698, 9.535282918202477427978936173487, 10.35123936646514334197028573270

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