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Properties

Label 2-117-117.49-c1-0-4
Degree $2$
Conductor $117$
Sign $0.886 - 0.462i$
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  + (0.339 + 0.195i)2-s + (1.10 + 1.33i)3-s + (−0.923 − 1.59i)4-s + (1.60 + 0.927i)5-s + (0.111 + 0.669i)6-s − 0.0822i·7-s − 1.50i·8-s + (−0.578 + 2.94i)9-s + (0.363 + 0.629i)10-s + (0.0464 + 0.0268i)11-s + (1.12 − 2.99i)12-s + (−3.60 − 0.0390i)13-s + (0.0161 − 0.0279i)14-s + (0.526 + 3.16i)15-s + (−1.55 + 2.68i)16-s + (2.16 − 3.75i)17-s + ⋯
L(s)  = 1  + (0.239 + 0.138i)2-s + (0.635 + 0.772i)3-s + (−0.461 − 0.799i)4-s + (0.718 + 0.414i)5-s + (0.0454 + 0.273i)6-s − 0.0310i·7-s − 0.532i·8-s + (−0.192 + 0.981i)9-s + (0.114 + 0.198i)10-s + (0.0140 + 0.00808i)11-s + (0.324 − 0.864i)12-s + (−0.999 − 0.0108i)13-s + (0.00430 − 0.00745i)14-s + (0.135 + 0.818i)15-s + (−0.387 + 0.671i)16-s + (0.525 − 0.909i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.886 - 0.462i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.886 - 0.462i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30617 + 0.319866i\)
\(L(\frac12)\) \(\approx\) \(1.30617 + 0.319866i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.10 - 1.33i)T \)
13 \( 1 + (3.60 + 0.0390i)T \)
good2 \( 1 + (-0.339 - 0.195i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.60 - 0.927i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 0.0822iT - 7T^{2} \)
11 \( 1 + (-0.0464 - 0.0268i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.16 + 3.75i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.45 + 2.57i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 1.58T + 23T^{2} \)
29 \( 1 + (-1.60 + 2.78i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.647 - 0.374i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.767 + 0.443i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 11.6iT - 41T^{2} \)
43 \( 1 + 5.79T + 43T^{2} \)
47 \( 1 + (8.48 - 4.89i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 - 12.5T + 53T^{2} \)
59 \( 1 + (-5.21 + 3.01i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 - 4.35iT - 67T^{2} \)
71 \( 1 + (-11.1 - 6.45i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 5.76iT - 73T^{2} \)
79 \( 1 + (5.17 + 8.96i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.68 - 5.59i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (9.76 - 5.63i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.1iT - 97T^{2} \)
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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

−13.87921502072775185290268055583, −13.02337106251196466882210093882, −11.28625368598762858627602313420, −9.953141005780632629603315815853, −9.814967165724885351815047498067, −8.480114227703532664481178369170, −6.84125600415525363587925715173, −5.43337553528932118058633052704, −4.42439703324691955241954610687, −2.56390641140704421769323793411, 2.17869925569543553370176759919, 3.76777980744129657699848693922, 5.43164706170529912981241860663, 6.98301869167342972159642845012, 8.179283783784849832390773214760, 8.942678307024129628225664406372, 10.11398198271161753787681507076, 11.91739510771771108419553543466, 12.64907845230791207896302607545, 13.27712435003379152826110324921

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