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Properties

Label 2-930-93.92-c1-0-37
Degree $2$
Conductor $930$
Sign $-0.977 - 0.210i$
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 + (1.58 + 0.687i)3-s − 4-s − i·5-s + (0.687 − 1.58i)6-s − 4.71·7-s + i·8-s + (2.05 + 2.18i)9-s − 10-s − 4.26·11-s + (−1.58 − 0.687i)12-s − 5.10i·13-s + 4.71i·14-s + (0.687 − 1.58i)15-s + 16-s + 2.21·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.917 + 0.396i)3-s − 0.5·4-s − 0.447i·5-s + (0.280 − 0.649i)6-s − 1.78·7-s + 0.353i·8-s + (0.685 + 0.728i)9-s − 0.316·10-s − 1.28·11-s + (−0.458 − 0.198i)12-s − 1.41i·13-s + 1.26i·14-s + (0.177 − 0.410i)15-s + 0.250·16-s + 0.537·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.977 - 0.210i$
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: \((2,\ 930,\ (\ :1/2),\ -0.977 - 0.210i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0471296 + 0.442038i\)
\(L(\frac12)\) \(\approx\) \(0.0471296 + 0.442038i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 - 0.687i)T \)
5 \( 1 + iT \)
31 \( 1 + (5.46 - 1.08i)T \)
good7 \( 1 + 4.71T + 7T^{2} \)
11 \( 1 + 4.26T + 11T^{2} \)
13 \( 1 + 5.10iT - 13T^{2} \)
17 \( 1 - 2.21T + 17T^{2} \)
19 \( 1 + 1.39T + 19T^{2} \)
23 \( 1 + 2.88T + 23T^{2} \)
29 \( 1 + 9.48T + 29T^{2} \)
37 \( 1 + 1.37iT - 37T^{2} \)
41 \( 1 + 9.55iT - 41T^{2} \)
43 \( 1 - 2.98iT - 43T^{2} \)
47 \( 1 - 5.85iT - 47T^{2} \)
53 \( 1 + 8.63T + 53T^{2} \)
59 \( 1 - 4.95iT - 59T^{2} \)
61 \( 1 + 7.20iT - 61T^{2} \)
67 \( 1 - 16.2T + 67T^{2} \)
71 \( 1 + 6.04iT - 71T^{2} \)
73 \( 1 + 12.2iT - 73T^{2} \)
79 \( 1 + 3.18iT - 79T^{2} \)
83 \( 1 - 3.17T + 83T^{2} \)
89 \( 1 + 5.58T + 89T^{2} \)
97 \( 1 - 2.28T + 97T^{2} \)
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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

−9.628636045262339388013190955404, −9.110399067580174666195225121785, −8.034411574418692560010435160640, −7.45037084606807599629937039011, −5.89629957741179527354203298430, −5.12795503402064740461014369329, −3.71343182800723732063431032429, −3.22192688629288444726832237258, −2.22521436583151046342049278271, −0.16995878354061720977922888542, 2.18842760352714814446544644175, 3.28889200508604513818049330616, 4.03488773561529314446242716255, 5.60193035674032900764079154906, 6.51385080024113713822357717270, 7.08682184774292800376533130610, 7.83785462767989032828178934926, 8.816610383006711737784401420214, 9.694972215508958526585968805651, 9.942791501643991516011500412823

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