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Properties

Label 2-117-117.43-c1-0-0
Degree $2$
Conductor $117$
Sign $-0.668 - 0.743i$
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  + (−2.24 + 1.29i)2-s + (−0.749 − 1.56i)3-s + (2.35 − 4.07i)4-s + (−1.18 + 0.685i)5-s + (3.70 + 2.53i)6-s + 3.70i·7-s + 7.01i·8-s + (−1.87 + 2.34i)9-s + (1.77 − 3.07i)10-s + (0.422 − 0.243i)11-s + (−8.13 − 0.618i)12-s + (−3.12 + 1.79i)13-s + (−4.79 − 8.31i)14-s + (1.96 + 1.33i)15-s + (−4.38 − 7.58i)16-s + (2.88 + 5.00i)17-s + ⋯
L(s)  = 1  + (−1.58 + 0.915i)2-s + (−0.432 − 0.901i)3-s + (1.17 − 2.03i)4-s + (−0.530 + 0.306i)5-s + (1.51 + 1.03i)6-s + 1.40i·7-s + 2.48i·8-s + (−0.625 + 0.780i)9-s + (0.561 − 0.972i)10-s + (0.127 − 0.0734i)11-s + (−2.34 − 0.178i)12-s + (−0.866 + 0.498i)13-s + (−1.28 − 2.22i)14-s + (0.506 + 0.345i)15-s + (−1.09 − 1.89i)16-s + (0.700 + 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.668 - 0.743i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.668 - 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.112345 + 0.251946i\)
\(L(\frac12)\) \(\approx\) \(0.112345 + 0.251946i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.749 + 1.56i)T \)
13 \( 1 + (3.12 - 1.79i)T \)
good2 \( 1 + (2.24 - 1.29i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (1.18 - 0.685i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.70iT - 7T^{2} \)
11 \( 1 + (-0.422 + 0.243i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.88 - 5.00i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.80 - 1.62i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.351T + 23T^{2} \)
29 \( 1 + (1.50 + 2.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.62 + 2.09i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.56 + 4.36i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 1.67iT - 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + (3.26 + 1.88i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 12.2T + 53T^{2} \)
59 \( 1 + (-4.94 - 2.85i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 2.35T + 61T^{2} \)
67 \( 1 - 0.0785iT - 67T^{2} \)
71 \( 1 + (-0.940 + 0.542i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 2.49iT - 73T^{2} \)
79 \( 1 + (1.16 - 2.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.19 - 1.26i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-14.0 - 8.08i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.846iT - 97T^{2} \)
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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

−14.43073430236596878613520717815, −12.54146969401186854733469057047, −11.69798029557385469801453612742, −10.65075014497289797775748523335, −9.347374653037339526585749931916, −8.313245473378694997033405713352, −7.57566126471358492521227268583, −6.41946026488886268973976794817, −5.59769306948010609304730762318, −2.02849496619887893480204130826, 0.51263880753600282412163044744, 3.23392431792815308839847606008, 4.60739596868638993780057150351, 7.03379550726172466203655709954, 8.006173243795687869806486249137, 9.255487662385939396237090850067, 10.11523973445137662497264846923, 10.74193416144112854945979894081, 11.70821062421097764202969982261, 12.52159888443664126605548568188

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