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Properties

Label 2-931-133.100-c1-0-41
Degree $2$
Conductor $931$
Sign $0.494 + 0.869i$
Analytic cond. $7.43407$
Root an. cond. $2.72654$
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  + (0.152 − 0.866i)2-s + (−0.5 + 0.181i)3-s + (1.15 + 0.419i)4-s + (2.37 − 0.866i)5-s + (0.0812 + 0.460i)6-s + (1.41 − 2.45i)8-s + (−2.08 + 1.74i)9-s + (−0.386 − 2.19i)10-s + 3.41·11-s − 0.652·12-s + (−0.918 − 5.21i)13-s + (−1.03 + 0.866i)15-s + (−0.0320 − 0.0269i)16-s + (1.26 + 1.06i)17-s + (1.19 + 2.06i)18-s + (−1.81 − 3.96i)19-s + ⋯
L(s)  = 1  + (0.107 − 0.612i)2-s + (−0.288 + 0.105i)3-s + (0.576 + 0.209i)4-s + (1.06 − 0.387i)5-s + (0.0331 + 0.188i)6-s + (0.501 − 0.868i)8-s + (−0.693 + 0.582i)9-s + (−0.122 − 0.693i)10-s + 1.02·11-s − 0.188·12-s + (−0.254 − 1.44i)13-s + (−0.266 + 0.223i)15-s + (−0.00802 − 0.00673i)16-s + (0.307 + 0.257i)17-s + (0.281 + 0.487i)18-s + (−0.417 − 0.908i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.494 + 0.869i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ 0.494 + 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91884 - 1.11597i\)
\(L(\frac12)\) \(\approx\) \(1.91884 - 1.11597i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 + (1.81 + 3.96i)T \)
good2 \( 1 + (-0.152 + 0.866i)T + (-1.87 - 0.684i)T^{2} \)
3 \( 1 + (0.5 - 0.181i)T + (2.29 - 1.92i)T^{2} \)
5 \( 1 + (-2.37 + 0.866i)T + (3.83 - 3.21i)T^{2} \)
11 \( 1 - 3.41T + 11T^{2} \)
13 \( 1 + (0.918 + 5.21i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-1.26 - 1.06i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-0.305 - 1.73i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-3.25 - 1.18i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (0.971 - 1.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.418 + 0.725i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.779 - 4.42i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-3.67 - 3.08i)T + (7.46 + 42.3i)T^{2} \)
47 \( 1 + (-0.549 + 0.460i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-5.73 - 2.08i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (8.24 + 6.91i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.762 - 4.32i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-2.46 - 13.9i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (10.5 + 8.84i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-7.06 + 2.57i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (5.33 + 4.47i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (1.25 + 2.17i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.14 - 0.780i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-1.71 + 0.623i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.15147420475789656713015486986, −9.286731732512654354998918979734, −8.349459743012626190351501962898, −7.35415279893573413521013906016, −6.28713949934110936054292849743, −5.63046689257844820708863692404, −4.63279842906843976087495331209, −3.25647040399714265114750755846, −2.37392580631929982877874809466, −1.17231938208277706127000141134, 1.57200204403479580959510099856, 2.54703263670113480522243379379, 4.03669401063858421425938259252, 5.33513155798511619731332611959, 6.24166472607217198999965346128, 6.44891404003075799614856941698, 7.32802313764934397147077813725, 8.588679125577964784706131167766, 9.364262718722016109999089115413, 10.16592926009865993392127699791

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