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Properties

Label 2-117-117.32-c1-0-9
Degree $2$
Conductor $117$
Sign $0.211 + 0.977i$
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  + (−0.662 − 0.662i)2-s + (1.56 − 0.749i)3-s − 1.12i·4-s + (0.101 − 0.0271i)5-s + (−1.53 − 0.538i)6-s + (−0.353 + 0.0946i)7-s + (−2.06 + 2.06i)8-s + (1.87 − 2.34i)9-s + (−0.0852 − 0.0492i)10-s + (−2.25 + 2.25i)11-s + (−0.840 − 1.75i)12-s + (3.51 − 0.821i)13-s + (0.296 + 0.171i)14-s + (0.137 − 0.118i)15-s + 0.499·16-s + (−0.713 − 1.23i)17-s + ⋯
L(s)  = 1  + (−0.468 − 0.468i)2-s + (0.901 − 0.432i)3-s − 0.560i·4-s + (0.0453 − 0.0121i)5-s + (−0.625 − 0.219i)6-s + (−0.133 + 0.0357i)7-s + (−0.731 + 0.731i)8-s + (0.625 − 0.780i)9-s + (−0.0269 − 0.0155i)10-s + (−0.678 + 0.678i)11-s + (−0.242 − 0.505i)12-s + (0.973 − 0.227i)13-s + (0.0793 + 0.0457i)14-s + (0.0356 − 0.0305i)15-s + 0.124·16-s + (−0.173 − 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.211 + 0.977i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.211 + 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.823874 - 0.664515i\)
\(L(\frac12)\) \(\approx\) \(0.823874 - 0.664515i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.56 + 0.749i)T \)
13 \( 1 + (-3.51 + 0.821i)T \)
good2 \( 1 + (0.662 + 0.662i)T + 2iT^{2} \)
5 \( 1 + (-0.101 + 0.0271i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.353 - 0.0946i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (2.25 - 2.25i)T - 11iT^{2} \)
17 \( 1 + (0.713 + 1.23i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.92 - 0.784i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.83 - 3.18i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5.17iT - 29T^{2} \)
31 \( 1 + (-1.46 - 5.46i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (4.32 - 1.15i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.76 + 6.57i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-1.94 - 1.12i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.42 + 1.18i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + 13.0iT - 53T^{2} \)
59 \( 1 + (-2.44 + 2.44i)T - 59iT^{2} \)
61 \( 1 + (-3.12 + 5.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (14.8 + 3.96i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.56 + 5.85i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (4.13 + 4.13i)T + 73iT^{2} \)
79 \( 1 + (-8.15 - 14.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.29 - 16.0i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (-0.630 - 2.35i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.20 + 11.9i)T + (-84.0 + 48.5i)T^{2} \)
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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

−13.39369475988908458893325690301, −12.31339220339387087354468075913, −11.08151574964284560613213791209, −9.976871801077293650207050038251, −9.176841166479062838011642880004, −8.119376900506964599941216339085, −6.84152733280874629132171007021, −5.32212392633173717968976939844, −3.24684859712359366844805849018, −1.65969914695201600667361128299, 2.89620233950953708193145692603, 4.15410005798290875439273172011, 6.12836780333993601457143143850, 7.59761945380543508043011898062, 8.369392802894357188810535220983, 9.221276917480413222463925729587, 10.34439429619991252067530937575, 11.62594151259679683072796938259, 13.12676230543906852521079970989, 13.62152925644921572891489284725

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