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Properties

Label 2-931-931.360-c1-0-2
Degree $2$
Conductor $931$
Sign $0.0194 - 0.999i$
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  + (−1.97 + 0.298i)4-s + (−2.83 − 3.05i)5-s + (−0.382 − 2.61i)7-s + (−2.47 − 1.68i)9-s + (−3.28 + 2.23i)11-s + (3.82 − 1.17i)16-s + (2.56 − 1.00i)17-s + (3.77 − 2.17i)19-s + (6.50 + 5.18i)20-s + (−3.00 + 7.64i)23-s + (−0.920 + 12.2i)25-s + (1.53 + 5.06i)28-s + (−6.90 + 8.57i)35-s + (5.40 + 2.60i)36-s + (2.78 − 12.2i)43-s + (5.82 − 5.40i)44-s + ⋯
L(s)  = 1  + (−0.988 + 0.149i)4-s + (−1.26 − 1.36i)5-s + (−0.144 − 0.989i)7-s + (−0.826 − 0.563i)9-s + (−0.989 + 0.674i)11-s + (0.955 − 0.294i)16-s + (0.622 − 0.244i)17-s + (0.866 − 0.499i)19-s + (1.45 + 1.16i)20-s + (−0.625 + 1.59i)23-s + (−0.184 + 2.45i)25-s + (0.290 + 0.956i)28-s + (−1.16 + 1.44i)35-s + (0.900 + 0.433i)36-s + (0.425 − 1.86i)43-s + (0.878 − 0.814i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0194 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0194 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $0.0194 - 0.999i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (360, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ 0.0194 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0575957 + 0.0564847i\)
\(L(\frac12)\) \(\approx\) \(0.0575957 + 0.0564847i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.382 + 2.61i)T \)
19 \( 1 + (-3.77 + 2.17i)T \)
good2 \( 1 + (1.97 - 0.298i)T^{2} \)
3 \( 1 + (2.47 + 1.68i)T^{2} \)
5 \( 1 + (2.83 + 3.05i)T + (-0.373 + 4.98i)T^{2} \)
11 \( 1 + (3.28 - 2.23i)T + (4.01 - 10.2i)T^{2} \)
13 \( 1 + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (-2.56 + 1.00i)T + (12.4 - 11.5i)T^{2} \)
23 \( 1 + (3.00 - 7.64i)T + (-16.8 - 15.6i)T^{2} \)
29 \( 1 + (6.45 + 28.2i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-35.3 - 10.9i)T^{2} \)
41 \( 1 + (-36.9 + 17.7i)T^{2} \)
43 \( 1 + (-2.78 + 12.2i)T + (-38.7 - 18.6i)T^{2} \)
47 \( 1 + (-0.332 + 0.0249i)T + (46.4 - 7.00i)T^{2} \)
53 \( 1 + (-50.6 + 15.6i)T^{2} \)
59 \( 1 + (4.40 + 58.8i)T^{2} \)
61 \( 1 + (-0.722 + 4.79i)T + (-58.2 - 17.9i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (15.7 - 69.2i)T^{2} \)
73 \( 1 + (16.0 + 1.20i)T + (72.1 + 10.8i)T^{2} \)
79 \( 1 + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.84 - 16.2i)T + (-51.7 + 64.8i)T^{2} \)
89 \( 1 + (32.5 + 82.8i)T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−10.01540454397310406613609654941, −9.349738991987850702734467319443, −8.587716583875973369997700484995, −7.72510163793210211575409568706, −7.40256966208060073563589625336, −5.49825012042695106010163629983, −4.97792490770295829745109898113, −3.98290526944995840013936095778, −3.37586857650097579060886747113, −0.946718612206606241255489819914, 0.05208946545079714822431265219, 2.72250475875598125526025448844, 3.26230349914704080193308938753, 4.47606520242468280656528786532, 5.57745342228388992940390377027, 6.22932160477287980914983867548, 7.71561651387733939395155370420, 8.090564715212059925943030425893, 8.797085251868488718757616867222, 10.06698498532495753777477779963

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