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Properties

Label 2-117-117.2-c1-0-3
Degree $2$
Conductor $117$
Sign $0.141 - 0.989i$
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  + (0.954 + 0.954i)2-s + (−0.157 + 1.72i)3-s − 0.177i·4-s + (−0.0632 + 0.236i)5-s + (−1.79 + 1.49i)6-s + (−0.804 + 3.00i)7-s + (2.07 − 2.07i)8-s + (−2.95 − 0.542i)9-s + (−0.285 + 0.165i)10-s + (4.14 − 4.14i)11-s + (0.305 + 0.0278i)12-s + (−3.46 − 1.00i)13-s + (−3.63 + 2.09i)14-s + (−0.397 − 0.146i)15-s + 3.61·16-s + (−0.931 + 1.61i)17-s + ⋯
L(s)  = 1  + (0.675 + 0.675i)2-s + (−0.0907 + 0.995i)3-s − 0.0885i·4-s + (−0.0282 + 0.105i)5-s + (−0.733 + 0.610i)6-s + (−0.304 + 1.13i)7-s + (0.734 − 0.734i)8-s + (−0.983 − 0.180i)9-s + (−0.0903 + 0.0521i)10-s + (1.24 − 1.24i)11-s + (0.0882 + 0.00804i)12-s + (−0.960 − 0.279i)13-s + (−0.971 + 0.560i)14-s + (−0.102 − 0.0377i)15-s + 0.903·16-s + (−0.226 + 0.391i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.141 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.141 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.141 - 0.989i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.141 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04356 + 0.905059i\)
\(L(\frac12)\) \(\approx\) \(1.04356 + 0.905059i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.157 - 1.72i)T \)
13 \( 1 + (3.46 + 1.00i)T \)
good2 \( 1 + (-0.954 - 0.954i)T + 2iT^{2} \)
5 \( 1 + (0.0632 - 0.236i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.804 - 3.00i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-4.14 + 4.14i)T - 11iT^{2} \)
17 \( 1 + (0.931 - 1.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.337 + 1.25i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.65 + 4.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.159iT - 29T^{2} \)
31 \( 1 + (7.81 + 2.09i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.98 - 7.40i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (8.02 - 2.15i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (3.08 - 1.78i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.23 - 4.62i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 - 2.20iT - 53T^{2} \)
59 \( 1 + (0.222 - 0.222i)T - 59iT^{2} \)
61 \( 1 + (4.08 + 7.08i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.14 - 8.01i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.12 - 1.10i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.634 + 0.634i)T + 73iT^{2} \)
79 \( 1 + (1.01 - 1.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.46 + 0.927i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-8.08 - 2.16i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (6.34 + 1.70i)T + (84.0 + 48.5i)T^{2} \)
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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

−14.21703804885922436858973938943, −12.89687695204087295611466364737, −11.69305327030848431218729198223, −10.65478183494266170041234873757, −9.440046613536209365924415723306, −8.618789909835384117789789407671, −6.65809993330832513608265609300, −5.76569634483782742341735385806, −4.75202419115324553074554744385, −3.26079696873218860204747975457, 1.86090903039995731574324162358, 3.64606946806041197372261982634, 4.94336143496394605546986234331, 6.97376066127913765475321241111, 7.37849558316385856058319395021, 9.064980624478992726129161532424, 10.48080788842100506296775510187, 11.68418943902739913412947051867, 12.32486215522814229707185589471, 13.10712321710675112090602207330

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