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Properties

Label 2-117-117.103-c1-0-4
Degree $2$
Conductor $117$
Sign $0.771 - 0.636i$
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  + (0.929 − 0.536i)2-s + (−0.744 + 1.56i)3-s + (−0.423 + 0.733i)4-s + (1.10 + 0.638i)5-s + (0.147 + 1.85i)6-s + (0.890 − 0.514i)7-s + 3.05i·8-s + (−1.89 − 2.32i)9-s + 1.37·10-s + (4.03 − 2.33i)11-s + (−0.831 − 1.20i)12-s + (−3.55 − 0.600i)13-s + (0.552 − 0.956i)14-s + (−1.82 + 1.25i)15-s + (0.794 + 1.37i)16-s − 0.476·17-s + ⋯
L(s)  = 1  + (0.657 − 0.379i)2-s + (−0.429 + 0.902i)3-s + (−0.211 + 0.366i)4-s + (0.494 + 0.285i)5-s + (0.0602 + 0.756i)6-s + (0.336 − 0.194i)7-s + 1.08i·8-s + (−0.630 − 0.776i)9-s + 0.433·10-s + (1.21 − 0.702i)11-s + (−0.240 − 0.348i)12-s + (−0.986 − 0.166i)13-s + (0.147 − 0.255i)14-s + (−0.470 + 0.323i)15-s + (0.198 + 0.344i)16-s − 0.115·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.771 - 0.636i$
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: \((2,\ 117,\ (\ :1/2),\ 0.771 - 0.636i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20633 + 0.433375i\)
\(L(\frac12)\) \(\approx\) \(1.20633 + 0.433375i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.744 - 1.56i)T \)
13 \( 1 + (3.55 + 0.600i)T \)
good2 \( 1 + (-0.929 + 0.536i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-1.10 - 0.638i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.890 + 0.514i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-4.03 + 2.33i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 0.476T + 17T^{2} \)
19 \( 1 + 6.69iT - 19T^{2} \)
23 \( 1 + (0.479 - 0.831i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.68 - 8.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.66 + 0.963i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.94iT - 37T^{2} \)
41 \( 1 + (1.31 + 0.762i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.31 + 2.27i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.92 + 3.41i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 0.582T + 53T^{2} \)
59 \( 1 + (3.64 + 2.10i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.71 + 8.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.01 + 1.16i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.35iT - 71T^{2} \)
73 \( 1 - 12.8iT - 73T^{2} \)
79 \( 1 + (-6.45 - 11.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.86 - 5.11i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.85iT - 89T^{2} \)
97 \( 1 + (14.9 - 8.63i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.88444556008821174014320691223, −12.45884294970336621258688813151, −11.57762518378553343521179552483, −10.79636793626305435456800426971, −9.510670098825505933562172484210, −8.561749284210607073210306662823, −6.74006124402528312508201936744, −5.29894853071200398795154367104, −4.31051208960541779924119032497, −2.96992804933466798400569947755, 1.70541425282673793179447651860, 4.39817172058245211016563440569, 5.60449158364151222936447319884, 6.46818003751130467503950903675, 7.63310837270463780683210258947, 9.217557160196622336169573774533, 10.21970918686108931105269749810, 11.86859253760402338474614991622, 12.42885652986915380619328296384, 13.53432731588115383151646854452

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