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Properties

Label 2-117-117.11-c1-0-9
Degree $2$
Conductor $117$
Sign $0.693 + 0.720i$
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  + (1.23 − 1.23i)2-s + (1.69 + 0.377i)3-s − 1.05i·4-s + (−1.93 − 0.519i)5-s + (2.55 − 1.62i)6-s + (−4.02 − 1.07i)7-s + (1.17 + 1.17i)8-s + (2.71 + 1.27i)9-s + (−3.03 + 1.75i)10-s + (1.90 + 1.90i)11-s + (0.397 − 1.77i)12-s + (−3.45 + 1.04i)13-s + (−6.30 + 3.63i)14-s + (−3.08 − 1.60i)15-s + 4.99·16-s + (2.33 − 4.05i)17-s + ⋯
L(s)  = 1  + (0.873 − 0.873i)2-s + (0.976 + 0.217i)3-s − 0.526i·4-s + (−0.867 − 0.232i)5-s + (1.04 − 0.662i)6-s + (−1.52 − 0.407i)7-s + (0.413 + 0.413i)8-s + (0.905 + 0.425i)9-s + (−0.960 + 0.554i)10-s + (0.573 + 0.573i)11-s + (0.114 − 0.513i)12-s + (−0.957 + 0.289i)13-s + (−1.68 + 0.972i)14-s + (−0.795 − 0.415i)15-s + 1.24·16-s + (0.567 − 0.982i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.693 + 0.720i$
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: \((2,\ 117,\ (\ :1/2),\ 0.693 + 0.720i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56662 - 0.666704i\)
\(L(\frac12)\) \(\approx\) \(1.56662 - 0.666704i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.69 - 0.377i)T \)
13 \( 1 + (3.45 - 1.04i)T \)
good2 \( 1 + (-1.23 + 1.23i)T - 2iT^{2} \)
5 \( 1 + (1.93 + 0.519i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (4.02 + 1.07i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.90 - 1.90i)T + 11iT^{2} \)
17 \( 1 + (-2.33 + 4.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.11 - 1.36i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.57 + 2.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.96iT - 29T^{2} \)
31 \( 1 + (-1.68 + 6.28i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-5.70 - 1.52i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.784 - 2.92i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.54 - 0.893i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.921 - 0.246i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 + (-0.106 - 0.106i)T + 59iT^{2} \)
61 \( 1 + (0.0799 + 0.138i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.65 - 1.78i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.98 - 7.39i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-5.30 + 5.30i)T - 73iT^{2} \)
79 \( 1 + (6.90 - 11.9i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.436 - 1.62i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (-3.29 + 12.3i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.22 - 8.29i)T + (-84.0 - 48.5i)T^{2} \)
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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

−13.12292071847235732777045397079, −12.59931404187609667960471437558, −11.68071589323908261066450139583, −10.21104954803752538439752770561, −9.452100506497182922695443465141, −7.975005577298229420456535063407, −6.87666555138199497092338609656, −4.57290760647452755508985305496, −3.77898823808768376477130195191, −2.63229966042129665427917706271, 3.16527601951065490503562382523, 4.12668561759686675069024841305, 6.02946779252003444741936382246, 6.93402777086265220995269342628, 7.943124642182472607231967561376, 9.240181751936809048972342870972, 10.34796920834322207982058884643, 12.22728321547597248000981607283, 12.86593063815809001631597388201, 13.78108838468885739640777935073

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