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Properties

Label 2-117-117.11-c1-0-8
Degree $2$
Conductor $117$
Sign $-0.862 + 0.506i$
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  + (−0.841 + 0.841i)2-s + (−0.947 − 1.44i)3-s + 0.583i·4-s + (−2.77 − 0.744i)5-s + (2.01 + 0.422i)6-s + (−1.42 − 0.381i)7-s + (−2.17 − 2.17i)8-s + (−1.20 + 2.74i)9-s + (2.96 − 1.71i)10-s + (−1.23 − 1.23i)11-s + (0.845 − 0.552i)12-s + (−3.60 + 0.117i)13-s + (1.52 − 0.878i)14-s + (1.55 + 4.73i)15-s + 2.49·16-s + (0.939 − 1.62i)17-s + ⋯
L(s)  = 1  + (−0.595 + 0.595i)2-s + (−0.547 − 0.837i)3-s + 0.291i·4-s + (−1.24 − 0.332i)5-s + (0.823 + 0.172i)6-s + (−0.538 − 0.144i)7-s + (−0.768 − 0.768i)8-s + (−0.401 + 0.915i)9-s + (0.937 − 0.541i)10-s + (−0.371 − 0.371i)11-s + (0.244 − 0.159i)12-s + (−0.999 + 0.0326i)13-s + (0.406 − 0.234i)14-s + (0.400 + 1.22i)15-s + 0.623·16-s + (0.227 − 0.394i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 + 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.862 + 0.506i$
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: \((2,\ 117,\ (\ :1/2),\ -0.862 + 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0209410 - 0.0770279i\)
\(L(\frac12)\) \(\approx\) \(0.0209410 - 0.0770279i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.947 + 1.44i)T \)
13 \( 1 + (3.60 - 0.117i)T \)
good2 \( 1 + (0.841 - 0.841i)T - 2iT^{2} \)
5 \( 1 + (2.77 + 0.744i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.42 + 0.381i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.23 + 1.23i)T + 11iT^{2} \)
17 \( 1 + (-0.939 + 1.62i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.34 + 1.16i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.82 - 4.89i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.70iT - 29T^{2} \)
31 \( 1 + (-0.676 + 2.52i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (11.1 + 2.99i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.168 - 0.627i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-1.78 + 1.02i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.31 + 0.619i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + 11.5iT - 53T^{2} \)
59 \( 1 + (9.15 + 9.15i)T + 59iT^{2} \)
61 \( 1 + (5.24 + 9.08i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.50 + 0.939i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-2.99 - 11.1i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (8.97 - 8.97i)T - 73iT^{2} \)
79 \( 1 + (4.85 - 8.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.712 - 2.65i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (1.57 - 5.86i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.35 - 5.06i)T + (-84.0 - 48.5i)T^{2} \)
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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

−12.78767969023492494652047486571, −12.15371966156919099486351742033, −11.34377191954787505579021288624, −9.717120299169237713550465037836, −8.372895857189301351860277819931, −7.52906651555751551329895658106, −6.92842575312944536699896366844, −5.28458397295547207289328018280, −3.37238881110457682841348591189, −0.10697566102603901021057465635, 3.07263713675853143227127558506, 4.61459376439171219711927255761, 6.02201400697550612230274411470, 7.59730882951197290457326364874, 8.991599100648537874947061396654, 10.08436203720587924428517533554, 10.59363070445858352340556792648, 11.92149208881601973024653726154, 12.13750920020695961205008014296, 14.25562590478839788475315691812

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