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Properties

Label 2-930-93.68-c1-0-43
Degree $2$
Conductor $930$
Sign $0.0488 - 0.998i$
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 + (−0.151 − 1.72i)3-s − 4-s + (0.866 + 0.5i)5-s + (−1.72 + 0.151i)6-s + (−1.64 − 2.85i)7-s + i·8-s + (−2.95 + 0.522i)9-s + (0.5 − 0.866i)10-s + (−2.34 + 4.05i)11-s + (0.151 + 1.72i)12-s + (−3.57 − 2.06i)13-s + (−2.85 + 1.64i)14-s + (0.731 − 1.56i)15-s + 16-s + (3.02 + 5.23i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.0874 − 0.996i)3-s − 0.5·4-s + (0.387 + 0.223i)5-s + (−0.704 + 0.0618i)6-s + (−0.622 − 1.07i)7-s + 0.353i·8-s + (−0.984 + 0.174i)9-s + (0.158 − 0.273i)10-s + (−0.706 + 1.22i)11-s + (0.0437 + 0.498i)12-s + (−0.991 − 0.572i)13-s + (−0.762 + 0.440i)14-s + (0.188 − 0.405i)15-s + 0.250·16-s + (0.732 + 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0488 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0488 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.0488 - 0.998i$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{930} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 0.0488 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0783206 + 0.0745868i\)
\(L(\frac12)\) \(\approx\) \(0.0783206 + 0.0745868i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.151 + 1.72i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (5.16 + 2.07i)T \)
good7 \( 1 + (1.64 + 2.85i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.34 - 4.05i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.57 + 2.06i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.02 - 5.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.11 + 1.93i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 0.244T + 23T^{2} \)
29 \( 1 + 2.05T + 29T^{2} \)
37 \( 1 + (7.39 - 4.27i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.83 + 3.94i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-10.0 + 5.79i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.99iT - 47T^{2} \)
53 \( 1 + (0.996 - 1.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-0.641 + 0.370i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 6.30iT - 61T^{2} \)
67 \( 1 + (-1.90 + 3.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.36 + 4.83i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.290 - 0.167i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (11.1 - 6.42i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.42 + 2.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.149T + 89T^{2} \)
97 \( 1 + 7.67T + 97T^{2} \)
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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

−9.770497372321864155807700781104, −8.595199168788589707707326111061, −7.42163824175202717353166941290, −7.22372435449030552352750320262, −5.96260109757720396557542228840, −5.02679850257588128278755199605, −3.74252864552856085984232818822, −2.62832111459531948707125992731, −1.64446063353012368700184683478, −0.04849908149598621570513162272, 2.59699409226469021143303327263, 3.51002570625187074674709289629, 4.92603151554678386639644912779, 5.48743742343809360045601837466, 6.09868543436206373623499831736, 7.31219765466141609296419870478, 8.428795442480363968203768413384, 9.101634830181089951026781496327, 9.625773597827496647813302749990, 10.41794744775266113777720396997

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