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Properties

Label 2-930-155.92-c1-0-16
Degree $2$
Conductor $930$
Sign $0.509 - 0.860i$
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 + (−0.707 − 0.707i)3-s + 1.00i·4-s + (1.89 − 1.19i)5-s − 1.00i·6-s + (2.48 + 2.48i)7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (2.18 + 0.496i)10-s + 3.08i·11-s + (0.707 − 0.707i)12-s + (−0.986 − 0.986i)13-s + 3.51i·14-s + (−2.18 − 0.496i)15-s − 1.00·16-s + (−1.61 + 1.61i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.408 − 0.408i)3-s + 0.500i·4-s + (0.846 − 0.532i)5-s − 0.408i·6-s + (0.938 + 0.938i)7-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (0.689 + 0.156i)10-s + 0.930i·11-s + (0.204 − 0.204i)12-s + (−0.273 − 0.273i)13-s + 0.938i·14-s + (−0.562 − 0.128i)15-s − 0.250·16-s + (−0.390 + 0.390i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.509 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.509 - 0.860i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.509 - 0.860i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93175 + 1.10067i\)
\(L(\frac12)\) \(\approx\) \(1.93175 + 1.10067i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (-1.89 + 1.19i)T \)
31 \( 1 + (-0.442 + 5.55i)T \)
good7 \( 1 + (-2.48 - 2.48i)T + 7iT^{2} \)
11 \( 1 - 3.08iT - 11T^{2} \)
13 \( 1 + (0.986 + 0.986i)T + 13iT^{2} \)
17 \( 1 + (1.61 - 1.61i)T - 17iT^{2} \)
19 \( 1 - 2.13iT - 19T^{2} \)
23 \( 1 + (-2.98 - 2.98i)T + 23iT^{2} \)
29 \( 1 - 3.68T + 29T^{2} \)
37 \( 1 + (0.671 - 0.671i)T - 37iT^{2} \)
41 \( 1 + 2.47T + 41T^{2} \)
43 \( 1 + (-1.90 - 1.90i)T + 43iT^{2} \)
47 \( 1 + (0.635 + 0.635i)T + 47iT^{2} \)
53 \( 1 + (-4.01 - 4.01i)T + 53iT^{2} \)
59 \( 1 - 7.25iT - 59T^{2} \)
61 \( 1 + 1.31iT - 61T^{2} \)
67 \( 1 + (0.962 + 0.962i)T + 67iT^{2} \)
71 \( 1 - 9.05T + 71T^{2} \)
73 \( 1 + (2.24 + 2.24i)T + 73iT^{2} \)
79 \( 1 - 14.6T + 79T^{2} \)
83 \( 1 + (7.83 + 7.83i)T + 83iT^{2} \)
89 \( 1 - 0.895T + 89T^{2} \)
97 \( 1 + (3.59 + 3.59i)T + 97iT^{2} \)
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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

−10.15476337204078311523926853265, −9.232800123438093960221978212276, −8.402897434712800711952909147148, −7.64136472425310855432210626459, −6.61233096017990662740965122050, −5.72875357334375087917020938404, −5.15481378135180442929033471568, −4.37075007564296764042850479748, −2.53755583012459903491890026443, −1.61665369075415974951331903866, 1.03817304991105830195518338861, 2.46164223269465360309560093716, 3.58236145578285707414587008908, 4.72015829694558448249971198097, 5.28306463160372536816871085636, 6.45143165489364550253824734972, 7.05990945601511796062566649129, 8.415188253655086775650933618414, 9.338087554582753851812175186133, 10.24953141688493919314786703754

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