Properties

Label 2-930-31.4-c1-0-18
Degree $2$
Conductor $930$
Sign $0.911 - 0.411i$
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  + (0.309 + 0.951i)2-s + (−0.309 + 0.951i)3-s + (−0.809 + 0.587i)4-s + 5-s − 0.999·6-s + (1.15 − 0.837i)7-s + (−0.809 − 0.587i)8-s + (−0.809 − 0.587i)9-s + (0.309 + 0.951i)10-s + (4.35 − 3.16i)11-s + (−0.309 − 0.951i)12-s + (2.13 − 6.56i)13-s + (1.15 + 0.837i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (0.246 + 0.179i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (−0.178 + 0.549i)3-s + (−0.404 + 0.293i)4-s + 0.447·5-s − 0.408·6-s + (0.435 − 0.316i)7-s + (−0.286 − 0.207i)8-s + (−0.269 − 0.195i)9-s + (0.0977 + 0.300i)10-s + (1.31 − 0.953i)11-s + (−0.0892 − 0.274i)12-s + (0.591 − 1.82i)13-s + (0.308 + 0.223i)14-s + (−0.0797 + 0.245i)15-s + (0.0772 − 0.237i)16-s + (0.0598 + 0.0434i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.911 - 0.411i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.911 - 0.411i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82752 + 0.393374i\)
\(L(\frac12)\) \(\approx\) \(1.82752 + 0.393374i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 - T \)
31 \( 1 + (3.97 - 3.89i)T \)
good7 \( 1 + (-1.15 + 0.837i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (-4.35 + 3.16i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-2.13 + 6.56i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.246 - 0.179i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.20 + 3.72i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.909 + 0.661i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.52 + 4.70i)T + (-23.4 + 17.0i)T^{2} \)
37 \( 1 - 4.52T + 37T^{2} \)
41 \( 1 + (-0.985 - 3.03i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-0.344 - 1.05i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (3.46 - 10.6i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.509 + 0.369i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.93 + 12.1i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 - 1.85T + 61T^{2} \)
67 \( 1 + 6.25T + 67T^{2} \)
71 \( 1 + (-11.5 - 8.40i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (13.0 - 9.45i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (12.7 + 9.28i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-4.31 - 13.2i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-10.6 + 7.73i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-5.07 + 3.69i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.05741303201146580105593026403, −9.174264822882313654291336679112, −8.447421741603059238726152265247, −7.66222685096468835732544527150, −6.39087898509270489327821196345, −5.89878438220730141351403753928, −4.95940168639589030427840299496, −3.94743431622977189061495129714, −3.02191993563212586335758271225, −0.944985133563604803090513383712, 1.61298095472237521362769035246, 1.97550393562791628741911943565, 3.74584826357178766240441445600, 4.52107907244601068525278703509, 5.70357535120753290635538751672, 6.52837189146723365800859728365, 7.30303566476071964589850077761, 8.663123787808629849024554091689, 9.185738540749537990109442581668, 9.999181881263967004866723416314

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