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Properties

Label 2-931-7.4-c1-0-57
Degree $2$
Conductor $931$
Sign $-0.991 + 0.126i$
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  + (−0.145 − 0.252i)2-s + (1.21 − 2.10i)3-s + (0.957 − 1.65i)4-s + (−1.03 − 1.78i)5-s − 0.708·6-s − 1.14·8-s + (−1.45 − 2.52i)9-s + (−0.300 + 0.521i)10-s + (2.47 − 4.28i)11-s + (−2.32 − 4.03i)12-s − 1.62·13-s − 5.02·15-s + (−1.74 − 3.02i)16-s + (−3.49 + 6.05i)17-s + (−0.424 + 0.735i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.103 − 0.178i)2-s + (0.702 − 1.21i)3-s + (0.478 − 0.829i)4-s + (−0.461 − 0.800i)5-s − 0.289·6-s − 0.403·8-s + (−0.485 − 0.841i)9-s + (−0.0951 + 0.164i)10-s + (0.746 − 1.29i)11-s + (−0.672 − 1.16i)12-s − 0.450·13-s − 1.29·15-s + (−0.437 − 0.757i)16-s + (−0.847 + 1.46i)17-s + (−0.100 + 0.173i)18-s + (0.114 + 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (704, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.117086 - 1.84503i\)
\(L(\frac12)\) \(\approx\) \(0.117086 - 1.84503i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
good2 \( 1 + (0.145 + 0.252i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.21 + 2.10i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.03 + 1.78i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.47 + 4.28i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.62T + 13T^{2} \)
17 \( 1 + (3.49 - 6.05i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-3.70 - 6.41i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.88T + 29T^{2} \)
31 \( 1 + (-0.404 + 0.700i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.05 + 3.55i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.50T + 41T^{2} \)
43 \( 1 - 7.31T + 43T^{2} \)
47 \( 1 + (2.70 + 4.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.995 + 1.72i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.87 - 4.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.69 + 9.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.99 - 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.51T + 71T^{2} \)
73 \( 1 + (1.04 - 1.80i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.15 - 2.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.01T + 83T^{2} \)
89 \( 1 + (-3.34 - 5.79i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14.0T + 97T^{2} \)
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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

−9.383698529604144102042106608398, −8.741279159876948445504606777126, −8.096202141499493903662254930508, −7.13288552046672607782793325736, −6.33192936085284294030740904662, −5.53792561811394589576614126328, −4.14618417698594205941351406428, −2.88418330966729358035634286946, −1.69167825582335714376461687086, −0.851145158389808579981530022399, 2.56404254780100823263837628123, 3.04164859741554940473563858519, 4.26765084662865382797752367773, 4.69801086378788115108981327640, 6.62955590709864897954700713180, 7.08040143274198047427008755975, 7.912953944415630876335071971775, 9.111266988124839011101481000047, 9.317493656335545016196775990389, 10.48658106745232365544385851477

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