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Properties

Label 2-117-117.86-c1-0-1
Degree $2$
Conductor $117$
Sign $0.762 - 0.646i$
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  + (−0.999 − 0.267i)2-s + (−1.73 − 0.0358i)3-s + (−0.805 − 0.465i)4-s + (0.483 + 1.80i)5-s + (1.72 + 0.499i)6-s + (3.99 + 1.07i)7-s + (2.14 + 2.14i)8-s + (2.99 + 0.124i)9-s − 1.93i·10-s + (−0.995 + 3.71i)11-s + (1.37 + 0.834i)12-s + (−2.74 − 2.34i)13-s + (−3.70 − 2.13i)14-s + (−0.772 − 3.14i)15-s + (−0.636 − 1.10i)16-s − 0.582·17-s + ⋯
L(s)  = 1  + (−0.706 − 0.189i)2-s + (−0.999 − 0.0206i)3-s + (−0.402 − 0.232i)4-s + (0.216 + 0.806i)5-s + (0.702 + 0.203i)6-s + (1.51 + 0.404i)7-s + (0.757 + 0.757i)8-s + (0.999 + 0.0413i)9-s − 0.610i·10-s + (−0.300 + 1.12i)11-s + (0.397 + 0.240i)12-s + (−0.760 − 0.649i)13-s + (−0.990 − 0.571i)14-s + (−0.199 − 0.810i)15-s + (−0.159 − 0.275i)16-s − 0.141·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.762 - 0.646i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.762 - 0.646i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.528918 + 0.194040i\)
\(L(\frac12)\) \(\approx\) \(0.528918 + 0.194040i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.73 + 0.0358i)T \)
13 \( 1 + (2.74 + 2.34i)T \)
good2 \( 1 + (0.999 + 0.267i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-0.483 - 1.80i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-3.99 - 1.07i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.995 - 3.71i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + 0.582T + 17T^{2} \)
19 \( 1 + (-4.41 - 4.41i)T + 19iT^{2} \)
23 \( 1 + (2.65 - 4.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.19 + 2.99i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.14 + 0.574i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (4.32 - 4.32i)T - 37iT^{2} \)
41 \( 1 + (1.64 + 6.13i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (5.20 - 3.00i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.64 + 6.15i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + 1.69iT - 53T^{2} \)
59 \( 1 + (-6.45 + 1.72i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.12 + 1.95i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.21 - 0.594i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.87 - 1.87i)T - 71iT^{2} \)
73 \( 1 + (-5.42 + 5.42i)T - 73iT^{2} \)
79 \( 1 + (1.10 + 1.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.0 + 2.95i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (0.110 + 0.110i)T + 89iT^{2} \)
97 \( 1 + (-2.91 + 10.8i)T + (-84.0 - 48.5i)T^{2} \)
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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

−13.76820543998169707102905968192, −12.20641199345490123406904442383, −11.45905833322914168981876075054, −10.26460056443369381401910932966, −9.952280471567240716682528293655, −8.166349017335637991936781490414, −7.27397957618144287980638175779, −5.51765095770429577542983445021, −4.73399641485754030290635031191, −1.81866309116641564446070128451, 1.01644226386091684264134987962, 4.52309152563571893428071425357, 5.15611134518996228982837190762, 6.98946390628369568458231613005, 8.127953177035622772053393454698, 9.030877673593137118347038839978, 10.29186001307263671663492586746, 11.26702168548702435512682195439, 12.23207122724305890939816612742, 13.39623203088095927854221194753

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