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Properties

Label 2-931-19.5-c1-0-19
Degree $2$
Conductor $931$
Sign $-0.600 - 0.799i$
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  + (−0.826 + 0.300i)2-s + (−0.0923 + 0.524i)3-s + (−0.939 + 0.788i)4-s + (1.93 + 1.62i)5-s + (−0.0812 − 0.460i)6-s + (1.41 − 2.45i)8-s + (2.55 + 0.929i)9-s + (−2.09 − 0.761i)10-s + (−1.70 + 2.95i)11-s + (−0.326 − 0.565i)12-s + (0.918 + 5.21i)13-s + (−1.03 + 0.866i)15-s + (−0.00727 + 0.0412i)16-s + (1.55 − 0.565i)17-s − 2.38·18-s + (2.52 − 3.55i)19-s + ⋯
L(s)  = 1  + (−0.584 + 0.212i)2-s + (−0.0533 + 0.302i)3-s + (−0.469 + 0.394i)4-s + (0.867 + 0.727i)5-s + (−0.0331 − 0.188i)6-s + (0.501 − 0.868i)8-s + (0.851 + 0.309i)9-s + (−0.661 − 0.240i)10-s + (−0.514 + 0.890i)11-s + (−0.0942 − 0.163i)12-s + (0.254 + 1.44i)13-s + (−0.266 + 0.223i)15-s + (−0.00181 + 0.0103i)16-s + (0.376 − 0.137i)17-s − 0.563·18-s + (0.578 − 0.815i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(931\)    =    \(7^{2} \cdot 19\)
Sign: $-0.600 - 0.799i$
Analytic conductor: \(7.43407\)
Root analytic conductor: \(2.72654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{931} (442, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 931,\ (\ :1/2),\ -0.600 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.531770 + 1.06421i\)
\(L(\frac12)\) \(\approx\) \(0.531770 + 1.06421i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.52 + 3.55i)T \)
good2 \( 1 + (0.826 - 0.300i)T + (1.53 - 1.28i)T^{2} \)
3 \( 1 + (0.0923 - 0.524i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (-1.93 - 1.62i)T + (0.868 + 4.92i)T^{2} \)
11 \( 1 + (1.70 - 2.95i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.918 - 5.21i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-1.55 + 0.565i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (-1.34 + 1.13i)T + (3.99 - 22.6i)T^{2} \)
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 \( 1 + 0.837T + 37T^{2} \)
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 \( 1 + (-0.673 - 0.245i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + (4.67 - 3.92i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (10.1 - 3.67i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (3.36 - 2.82i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (13.3 + 4.86i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (10.5 + 8.84i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-1.30 + 7.40i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (1.20 - 6.85i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-1.25 - 2.17i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.396 - 2.24i)T + (-83.6 + 30.4i)T^{2} \)
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

−10.26136901037696907074757087004, −9.402328693400401782199099009906, −9.059736886455392584785763660645, −7.66725843543971893851598439963, −7.13591477951985617644395281365, −6.34488013058754632188260380767, −4.89334221045397939209575772186, −4.30666263801043232246375248473, −2.91609285913011192712331119106, −1.61272382650658084868150760851, 0.78225425341001874425856185193, 1.58454305839813082220613907621, 3.16705985891640804780148891557, 4.61290265571066541782161557918, 5.62152299207878350454652985738, 5.96138863935894336208535857763, 7.56693952059923563281442272129, 8.219572829675654130980033058141, 9.014767098305219187368894614442, 9.913733775474942820304491024134

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