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Properties

Label 2-117-117.11-c1-0-4
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: $\chi_{117} (11, \cdot )$
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

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

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