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Properties

Label 2-117-117.47-c1-0-9
Degree $2$
Conductor $117$
Sign $-0.662 + 0.748i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
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  + (−0.535 − 1.99i)2-s + (1.67 − 0.437i)3-s + (−1.97 + 1.13i)4-s + (−2.39 − 0.641i)5-s + (−1.77 − 3.11i)6-s + (−0.185 − 0.693i)7-s + (0.403 + 0.403i)8-s + (2.61 − 1.46i)9-s + 5.12i·10-s + (2.71 − 0.728i)11-s + (−2.80 + 2.76i)12-s + (2.59 + 2.50i)13-s + (−1.28 + 0.742i)14-s + (−4.29 − 0.0276i)15-s + (−1.68 + 2.92i)16-s − 1.67·17-s + ⋯
L(s)  = 1  + (−0.378 − 1.41i)2-s + (0.967 − 0.252i)3-s + (−0.985 + 0.568i)4-s + (−1.07 − 0.287i)5-s + (−0.722 − 1.27i)6-s + (−0.0702 − 0.262i)7-s + (0.142 + 0.142i)8-s + (0.872 − 0.488i)9-s + 1.62i·10-s + (0.819 − 0.219i)11-s + (−0.809 + 0.799i)12-s + (0.720 + 0.693i)13-s + (−0.343 + 0.198i)14-s + (−1.10 − 0.00714i)15-s + (−0.421 + 0.730i)16-s − 0.406·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.662 + 0.748i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.662 + 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.415495 - 0.922641i\)
\(L(\frac12)\) \(\approx\) \(0.415495 - 0.922641i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 + 0.437i)T \)
13 \( 1 + (-2.59 - 2.50i)T \)
good2 \( 1 + (0.535 + 1.99i)T + (-1.73 + i)T^{2} \)
5 \( 1 + (2.39 + 0.641i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.185 + 0.693i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.71 + 0.728i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 1.67T + 17T^{2} \)
19 \( 1 + (-1.83 - 1.83i)T + 19iT^{2} \)
23 \( 1 + (-4.29 - 7.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.554 + 0.320i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.00 + 7.49i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (6.11 - 6.11i)T - 37iT^{2} \)
41 \( 1 + (0.775 + 0.207i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.31 + 3.06i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (12.4 - 3.34i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 - 2.05iT - 53T^{2} \)
59 \( 1 + (-0.892 + 3.33i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.19 + 5.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.781 - 2.91i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (10.3 - 10.3i)T - 71iT^{2} \)
73 \( 1 + (2.15 - 2.15i)T - 73iT^{2} \)
79 \( 1 + (5.01 - 8.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.35 - 5.07i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-5.67 - 5.67i)T + 89iT^{2} \)
97 \( 1 + (-16.2 + 4.35i)T + (84.0 - 48.5i)T^{2} \)
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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

−13.04386679917146028573541846692, −11.77452466916881841811657419523, −11.43440753923349962735556533930, −9.908773258962404127612838297234, −9.034878350999756952521294608707, −8.160378303189633731203505572806, −6.81354562736902763729577800585, −4.09170919454478396815392347925, −3.37988914632170924871102458302, −1.45543121947333423517004592671, 3.26687092369322315543833475214, 4.77877228501896496900471409090, 6.58807260032070142221710495431, 7.43363596654913298535088542413, 8.499869100871637831400590321873, 9.003701774216804545946845483247, 10.53851923980565499302829298184, 11.89200942542578506542145879477, 13.25855922344210642944494293408, 14.50701905683177700721058039010

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