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Properties

Label 2-930-93.92-c1-0-28
Degree $2$
Conductor $930$
Sign $0.560 + 0.828i$
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 + (−0.224 + 1.71i)3-s − 4-s + i·5-s + (−1.71 − 0.224i)6-s − 2.17·7-s − i·8-s + (−2.89 − 0.770i)9-s − 10-s − 3.24·11-s + (0.224 − 1.71i)12-s − 4.02i·13-s − 2.17i·14-s + (−1.71 − 0.224i)15-s + 16-s + 1.33·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.129 + 0.991i)3-s − 0.5·4-s + 0.447i·5-s + (−0.701 − 0.0915i)6-s − 0.820·7-s − 0.353i·8-s + (−0.966 − 0.256i)9-s − 0.316·10-s − 0.977·11-s + (0.0647 − 0.495i)12-s − 1.11i·13-s − 0.580i·14-s + (−0.443 − 0.0579i)15-s + 0.250·16-s + 0.324·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.560 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.560 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.560 + 0.828i$
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: \((2,\ 930,\ (\ :1/2),\ 0.560 + 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.151896 - 0.0806168i\)
\(L(\frac12)\) \(\approx\) \(0.151896 - 0.0806168i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.224 - 1.71i)T \)
5 \( 1 - iT \)
31 \( 1 + (-4.16 + 3.69i)T \)
good7 \( 1 + 2.17T + 7T^{2} \)
11 \( 1 + 3.24T + 11T^{2} \)
13 \( 1 + 4.02iT - 13T^{2} \)
17 \( 1 - 1.33T + 17T^{2} \)
19 \( 1 - 5.97T + 19T^{2} \)
23 \( 1 + 6.67T + 23T^{2} \)
29 \( 1 + 4.69T + 29T^{2} \)
37 \( 1 + 3.43iT - 37T^{2} \)
41 \( 1 + 5.45iT - 41T^{2} \)
43 \( 1 - 7.95iT - 43T^{2} \)
47 \( 1 - 11.1iT - 47T^{2} \)
53 \( 1 + 11.9T + 53T^{2} \)
59 \( 1 + 12.6iT - 59T^{2} \)
61 \( 1 - 3.87iT - 61T^{2} \)
67 \( 1 + 3.59T + 67T^{2} \)
71 \( 1 - 8.31iT - 71T^{2} \)
73 \( 1 + 14.2iT - 73T^{2} \)
79 \( 1 + 12.4iT - 79T^{2} \)
83 \( 1 + 0.448T + 83T^{2} \)
89 \( 1 + 5.85T + 89T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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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

−9.828299138115802241840786172955, −9.372218090830853929728845707743, −8.003842480076284729506466434240, −7.64046006686913683740786316926, −6.19054315610122869935702426318, −5.72237473785508175428665216870, −4.79217831634352924044217712586, −3.55697582680666832673083277709, −2.87696725914666727273695595428, −0.082246094085917184323013618597, 1.45023260639379969472640372323, 2.58873392691082040626790815292, 3.63188418698938980237088171253, 4.99827933378062693552827837017, 5.83105917981129889021403796449, 6.83072898162274880335948644144, 7.75306635443553525142793699825, 8.517887176208275581946918741656, 9.505606701691274396273342670787, 10.10972385355000057941249301966

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