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Properties

Label 2-117-117.110-c1-0-8
Degree $2$
Conductor $117$
Sign $0.168 + 0.985i$
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  + (0.242 − 0.905i)2-s + (−0.131 − 1.72i)3-s + (0.971 + 0.560i)4-s + (0.0271 − 0.101i)5-s + (−1.59 − 0.299i)6-s + (0.258 + 0.258i)7-s + (2.06 − 2.06i)8-s + (−2.96 + 0.455i)9-s + (−0.0852 − 0.0492i)10-s + (−3.07 − 0.824i)11-s + (0.840 − 1.75i)12-s + (−1.04 + 3.45i)13-s + (0.296 − 0.171i)14-s + (−0.178 − 0.0335i)15-s + (−0.249 − 0.432i)16-s + (0.713 + 1.23i)17-s + ⋯
L(s)  = 1  + (0.171 − 0.640i)2-s + (−0.0760 − 0.997i)3-s + (0.485 + 0.280i)4-s + (0.0121 − 0.0453i)5-s + (−0.651 − 0.122i)6-s + (0.0976 + 0.0976i)7-s + (0.731 − 0.731i)8-s + (−0.988 + 0.151i)9-s + (−0.0269 − 0.0155i)10-s + (−0.927 − 0.248i)11-s + (0.242 − 0.505i)12-s + (−0.289 + 0.957i)13-s + (0.0793 − 0.0457i)14-s + (−0.0461 − 0.00866i)15-s + (−0.0623 − 0.108i)16-s + (0.173 + 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.168 + 0.985i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.168 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.938166 - 0.791731i\)
\(L(\frac12)\) \(\approx\) \(0.938166 - 0.791731i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.131 + 1.72i)T \)
13 \( 1 + (1.04 - 3.45i)T \)
good2 \( 1 + (-0.242 + 0.905i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-0.0271 + 0.101i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.258 - 0.258i)T + 7iT^{2} \)
11 \( 1 + (3.07 + 0.824i)T + (9.52 + 5.5i)T^{2} \)
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 \( 1 - 3.67T + 23T^{2} \)
29 \( 1 + (4.47 - 2.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.46 + 1.46i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (4.32 - 1.15i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (4.81 + 4.81i)T + 41iT^{2} \)
43 \( 1 + 2.24iT - 43T^{2} \)
47 \( 1 + (1.18 + 4.42i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 13.0iT - 53T^{2} \)
59 \( 1 + (0.895 + 3.34i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + 6.24T + 61T^{2} \)
67 \( 1 + (-10.8 + 10.8i)T - 67iT^{2} \)
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 \( 1 + (16.0 - 4.29i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (0.630 + 2.35i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.74 - 8.74i)T - 97iT^{2} \)
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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

−13.04820790399069450679022547067, −12.30608738808915054475420588244, −11.41003789605453781631718989887, −10.58776052037416308045186008347, −8.962977732311185084143280708044, −7.62387594259986648069176291827, −6.86594775538008002798497336694, −5.31728767678406478891899167259, −3.26212923021402210847367376198, −1.84046907380049184830549429227, 2.92516376887372025462030309768, 4.90872170241101730754109889191, 5.60970129255875689673502736396, 7.14618261126607216154178189693, 8.235916541146867036787847226250, 9.731313443046683122832648514958, 10.62046369397901079674314732031, 11.38348961702003075780629520018, 12.85678593373932720338573930170, 14.17620917113977684199822711996

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