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Properties

Label 2-117-117.5-c1-0-6
Degree $2$
Conductor $117$
Sign $0.947 + 0.320i$
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  + (−0.120 + 0.449i)2-s + (−0.453 − 1.67i)3-s + (1.54 + 0.891i)4-s + (0.207 − 0.0555i)5-s + (0.805 − 0.00273i)6-s + (0.845 − 3.15i)7-s + (−1.24 + 1.24i)8-s + (−2.58 + 1.51i)9-s + 0.0998i·10-s + (3.19 + 0.856i)11-s + (0.789 − 2.98i)12-s + (3.49 − 0.877i)13-s + (1.31 + 0.758i)14-s + (−0.187 − 0.321i)15-s + (1.37 + 2.38i)16-s − 7.72·17-s + ⋯
L(s)  = 1  + (−0.0850 + 0.317i)2-s + (−0.262 − 0.965i)3-s + (0.772 + 0.445i)4-s + (0.0927 − 0.0248i)5-s + (0.328 − 0.00111i)6-s + (0.319 − 1.19i)7-s + (−0.439 + 0.439i)8-s + (−0.862 + 0.505i)9-s + 0.0315i·10-s + (0.964 + 0.258i)11-s + (0.227 − 0.862i)12-s + (0.969 − 0.243i)13-s + (0.351 + 0.202i)14-s + (−0.0482 − 0.0829i)15-s + (0.343 + 0.595i)16-s − 1.87·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.947 + 0.320i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.947 + 0.320i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09034 - 0.179295i\)
\(L(\frac12)\) \(\approx\) \(1.09034 - 0.179295i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.453 + 1.67i)T \)
13 \( 1 + (-3.49 + 0.877i)T \)
good2 \( 1 + (0.120 - 0.449i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (-0.207 + 0.0555i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.845 + 3.15i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3.19 - 0.856i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 7.72T + 17T^{2} \)
19 \( 1 + (3.13 - 3.13i)T - 19iT^{2} \)
23 \( 1 + (-0.172 + 0.298i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.06 - 2.34i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.173 + 0.647i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.04 + 1.04i)T + 37iT^{2} \)
41 \( 1 + (-7.17 + 1.92i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.27 - 1.88i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-10.6 - 2.85i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + 1.94iT - 53T^{2} \)
59 \( 1 + (2.48 + 9.27i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (3.25 + 5.63i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.729 - 2.72i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-2.22 - 2.22i)T + 71iT^{2} \)
73 \( 1 + (1.50 + 1.50i)T + 73iT^{2} \)
79 \( 1 + (-3.82 - 6.63i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.87 + 14.4i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (1.95 - 1.95i)T - 89iT^{2} \)
97 \( 1 + (2.62 + 0.702i)T + (84.0 + 48.5i)T^{2} \)
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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

−13.40896508449540036750228474605, −12.52896743587850705084207056040, −11.27751523043862687302073735129, −10.86221907588711192638201240085, −8.856021258551706119101680545219, −7.76711092248940051038774420239, −6.87288364230842085480291216591, −6.07728132180443141843911538648, −3.94985345854384215995662356958, −1.82177704560490068289962395524, 2.34554837995623654593870732644, 4.14819662295817992110994329727, 5.77849372629995114720117949304, 6.51658740895732936259561241941, 8.762303865397091093257421713693, 9.280522139218420037471758327473, 10.74925348316307730337527186470, 11.33791595983954137673317572971, 12.07313378907519767849831207589, 13.69565896499399904241309285968

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