Properties

Label 2-930-15.2-c1-0-0
Degree $2$
Conductor $930$
Sign $-0.295 + 0.955i$
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  + (−0.707 + 0.707i)2-s + (0.0980 + 1.72i)3-s − 1.00i·4-s + (−0.278 + 2.21i)5-s + (−1.29 − 1.15i)6-s + (−1.59 − 1.59i)7-s + (0.707 + 0.707i)8-s + (−2.98 + 0.339i)9-s + (−1.37 − 1.76i)10-s + 3.70i·11-s + (1.72 − 0.0980i)12-s + (−1.02 + 1.02i)13-s + 2.25·14-s + (−3.86 − 0.264i)15-s − 1.00·16-s + (0.479 − 0.479i)17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (0.0566 + 0.998i)3-s − 0.500i·4-s + (−0.124 + 0.992i)5-s + (−0.527 − 0.470i)6-s + (−0.602 − 0.602i)7-s + (0.250 + 0.250i)8-s + (−0.993 + 0.113i)9-s + (−0.433 − 0.558i)10-s + 1.11i·11-s + (0.499 − 0.0283i)12-s + (−0.284 + 0.284i)13-s + 0.602·14-s + (−0.997 − 0.0682i)15-s − 0.250·16-s + (0.116 − 0.116i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.295 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.295 + 0.955i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ -0.295 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.200164 - 0.271484i\)
\(L(\frac12)\) \(\approx\) \(0.200164 - 0.271484i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (-0.0980 - 1.72i)T \)
5 \( 1 + (0.278 - 2.21i)T \)
31 \( 1 + T \)
good7 \( 1 + (1.59 + 1.59i)T + 7iT^{2} \)
11 \( 1 - 3.70iT - 11T^{2} \)
13 \( 1 + (1.02 - 1.02i)T - 13iT^{2} \)
17 \( 1 + (-0.479 + 0.479i)T - 17iT^{2} \)
19 \( 1 + 0.296iT - 19T^{2} \)
23 \( 1 + (1.49 + 1.49i)T + 23iT^{2} \)
29 \( 1 + 1.30T + 29T^{2} \)
37 \( 1 + (2.24 + 2.24i)T + 37iT^{2} \)
41 \( 1 + 5.01iT - 41T^{2} \)
43 \( 1 + (7.37 - 7.37i)T - 43iT^{2} \)
47 \( 1 + (0.230 - 0.230i)T - 47iT^{2} \)
53 \( 1 + (0.282 + 0.282i)T + 53iT^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 + (9.42 + 9.42i)T + 67iT^{2} \)
71 \( 1 - 6.43iT - 71T^{2} \)
73 \( 1 + (5.91 - 5.91i)T - 73iT^{2} \)
79 \( 1 + 3.56iT - 79T^{2} \)
83 \( 1 + (1.95 + 1.95i)T + 83iT^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + (-5.06 - 5.06i)T + 97iT^{2} \)
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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

−10.18900009138952484133149357125, −10.02848644614192173729893250355, −9.196465458646355154525366294034, −8.119250409315158598555781954089, −7.16960750179731589857580943092, −6.63060210544324341439634685276, −5.51189641906409409362509490713, −4.41794752869720112777922015709, −3.54169046748048631187663183164, −2.30647945238761732615322848890, 0.18444978821250284816287838923, 1.45225022268877060210336903786, 2.70933795505514056167974851940, 3.69890052251244771574386974244, 5.29822606296862626167790814900, 5.99923129386236256741803430434, 7.07956864021863832606506051967, 8.128868944911123185795272513491, 8.564547628616181447681122939989, 9.292409345585814801633061148637

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