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Properties

Label 2-117-117.103-c1-0-7
Degree $2$
Conductor $117$
Sign $0.908 + 0.418i$
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  + (0.784 − 0.452i)2-s + (1.66 − 0.460i)3-s + (−0.589 + 1.02i)4-s + (−1.94 − 1.12i)5-s + (1.10 − 1.11i)6-s + (2.97 − 1.71i)7-s + 2.87i·8-s + (2.57 − 1.53i)9-s − 2.03·10-s + (−3.20 + 1.84i)11-s + (−0.514 + 1.97i)12-s + (−3.28 + 1.48i)13-s + (1.55 − 2.69i)14-s + (−3.76 − 0.977i)15-s + (0.124 + 0.214i)16-s − 4.21·17-s + ⋯
L(s)  = 1  + (0.554 − 0.320i)2-s + (0.963 − 0.265i)3-s + (−0.294 + 0.510i)4-s + (−0.868 − 0.501i)5-s + (0.449 − 0.456i)6-s + (1.12 − 0.649i)7-s + 1.01i·8-s + (0.858 − 0.512i)9-s − 0.642·10-s + (−0.965 + 0.557i)11-s + (−0.148 + 0.570i)12-s + (−0.910 + 0.413i)13-s + (0.415 − 0.720i)14-s + (−0.970 − 0.252i)15-s + (0.0310 + 0.0537i)16-s − 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.908 + 0.418i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.908 + 0.418i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48292 - 0.325357i\)
\(L(\frac12)\) \(\approx\) \(1.48292 - 0.325357i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.66 + 0.460i)T \)
13 \( 1 + (3.28 - 1.48i)T \)
good2 \( 1 + (-0.784 + 0.452i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.94 + 1.12i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.97 + 1.71i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.20 - 1.84i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 4.21T + 17T^{2} \)
19 \( 1 - 4.25iT - 19T^{2} \)
23 \( 1 + (-1.89 + 3.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.18 + 2.06i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.37 - 3.67i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.49iT - 37T^{2} \)
41 \( 1 + (-6.86 - 3.96i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.450 + 0.779i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.80 + 2.77i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.59T + 53T^{2} \)
59 \( 1 + (4.44 + 2.56i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.50 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.7 + 6.75i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.65iT - 71T^{2} \)
73 \( 1 + 5.45iT - 73T^{2} \)
79 \( 1 + (5.46 + 9.47i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.465 - 0.268i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 5.75iT - 89T^{2} \)
97 \( 1 + (-5.87 + 3.39i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.42627247803761415975972527718, −12.51304471726831427597845625483, −11.80354881540335162951658748590, −10.45065909562141183213618742308, −8.851343620266959351494513723614, −7.967385261493788168011209610831, −7.40507805579315695471862184821, −4.69349640861510406987135469596, −4.17447821097379105194242146474, −2.38970374191528453158869914150, 2.72063594259214437631451158750, 4.37421326920522768615630431592, 5.30845226870117612093797211820, 7.18007087881966261940765590632, 8.136392978356016306010609306072, 9.208223670905476728490969442864, 10.52109119705162052315311696805, 11.46344563742377503889894300805, 12.97235978253983439651759927752, 13.79528241282063919456911280285

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