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Properties

Label 2-930-155.92-c1-0-27
Degree $2$
Conductor $930$
Sign $0.111 + 0.993i$
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 + (−1.96 − 1.06i)5-s + 1.00i·6-s + (−1.95 − 1.95i)7-s + (−0.707 + 0.707i)8-s + 1.00i·9-s + (−0.633 − 2.14i)10-s − 0.865i·11-s + (−0.707 + 0.707i)12-s + (−3.63 − 3.63i)13-s − 2.76i·14-s + (−0.633 − 2.14i)15-s − 1.00·16-s + (2.10 − 2.10i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.408 + 0.408i)3-s + 0.500i·4-s + (−0.878 − 0.477i)5-s + 0.408i·6-s + (−0.738 − 0.738i)7-s + (−0.250 + 0.250i)8-s + 0.333i·9-s + (−0.200 − 0.678i)10-s − 0.260i·11-s + (−0.204 + 0.204i)12-s + (−1.00 − 1.00i)13-s − 0.738i·14-s + (−0.163 − 0.553i)15-s − 0.250·16-s + (0.510 − 0.510i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.111 + 0.993i$
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: \((2,\ 930,\ (\ :1/2),\ 0.111 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.686878 - 0.613886i\)
\(L(\frac12)\) \(\approx\) \(0.686878 - 0.613886i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.96 + 1.06i)T \)
31 \( 1 + (-5.55 + 0.316i)T \)
good7 \( 1 + (1.95 + 1.95i)T + 7iT^{2} \)
11 \( 1 + 0.865iT - 11T^{2} \)
13 \( 1 + (3.63 + 3.63i)T + 13iT^{2} \)
17 \( 1 + (-2.10 + 2.10i)T - 17iT^{2} \)
19 \( 1 - 0.565iT - 19T^{2} \)
23 \( 1 + (6.50 + 6.50i)T + 23iT^{2} \)
29 \( 1 - 0.232T + 29T^{2} \)
37 \( 1 + (-2.57 + 2.57i)T - 37iT^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 + (4.95 + 4.95i)T + 43iT^{2} \)
47 \( 1 + (-4.14 - 4.14i)T + 47iT^{2} \)
53 \( 1 + (0.513 + 0.513i)T + 53iT^{2} \)
59 \( 1 - 4.60iT - 59T^{2} \)
61 \( 1 - 2.89iT - 61T^{2} \)
67 \( 1 + (1.09 + 1.09i)T + 67iT^{2} \)
71 \( 1 + 9.81T + 71T^{2} \)
73 \( 1 + (0.542 + 0.542i)T + 73iT^{2} \)
79 \( 1 - 9.17T + 79T^{2} \)
83 \( 1 + (0.433 + 0.433i)T + 83iT^{2} \)
89 \( 1 + 3.35T + 89T^{2} \)
97 \( 1 + (-6.46 - 6.46i)T + 97iT^{2} \)
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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

−10.01826332573983948145409589757, −8.837979201190901945940094379927, −8.029339042319289617859559432989, −7.47435907801265371250131417216, −6.50223669425325514449856350992, −5.31697957900426825558833890333, −4.47273238252540904297691615614, −3.65392952640693688373060017399, −2.78432204362839721742954737092, −0.33099784034745582870538124327, 1.88566190661525682970387243129, 2.93138363320087238702226059298, 3.74012849499296630684874805751, 4.78334782862415576726589916463, 6.05480598039755779560574560368, 6.80766721086192130826737000691, 7.68394038043644627573547282180, 8.579771423084285250176320471562, 9.677419496066226351550765231654, 10.07997845756361762662927974552

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