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Properties

Label 2-117-117.5-c1-0-8
Degree 22
Conductor 117117
Sign 0.713+0.700i0.713 + 0.700i
Analytic cond. 0.9342490.934249
Root an. cond. 0.9665650.966565
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.407 − 1.51i)2-s + (1.33 + 1.10i)3-s + (−0.412 − 0.237i)4-s + (−1.74 + 0.466i)5-s + (2.21 − 1.58i)6-s + (0.556 − 2.07i)7-s + (1.69 − 1.69i)8-s + (0.575 + 2.94i)9-s + 2.83i·10-s + (−2.79 − 0.748i)11-s + (−0.289 − 0.771i)12-s + (0.478 + 3.57i)13-s + (−2.93 − 1.69i)14-s + (−2.84 − 1.29i)15-s + (−2.36 − 4.09i)16-s − 6.33·17-s + ⋯
L(s)  = 1  + (0.287 − 1.07i)2-s + (0.771 + 0.635i)3-s + (−0.206 − 0.118i)4-s + (−0.778 + 0.208i)5-s + (0.905 − 0.646i)6-s + (0.210 − 0.785i)7-s + (0.599 − 0.599i)8-s + (0.191 + 0.981i)9-s + 0.896i·10-s + (−0.842 − 0.225i)11-s + (−0.0834 − 0.222i)12-s + (0.132 + 0.991i)13-s + (−0.783 − 0.452i)14-s + (−0.733 − 0.333i)15-s + (−0.590 − 1.02i)16-s − 1.53·17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.713+0.700i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((0.713+0.700i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.713+0.700i0.713 + 0.700i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ117(5,⋅)\chi_{117} (5, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1/2), 0.713+0.700i)(2,\ 117,\ (\ :1/2),\ 0.713 + 0.700i)

Particular Values

L(1)L(1) ≈\approx 1.30742−0.534441i1.30742 - 0.534441i
L(12)L(\frac12) ≈\approx 1.30742−0.534441i1.30742 - 0.534441i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(−1.33−1.10i)T 1 + (-1.33 - 1.10i)T
13 1+(−0.478−3.57i)T 1 + (-0.478 - 3.57i)T
good2 1+(−0.407+1.51i)T+(−1.73−i)T2 1 + (-0.407 + 1.51i)T + (-1.73 - i)T^{2}
5 1+(1.74−0.466i)T+(4.33−2.5i)T2 1 + (1.74 - 0.466i)T + (4.33 - 2.5i)T^{2}
7 1+(−0.556+2.07i)T+(−6.06−3.5i)T2 1 + (-0.556 + 2.07i)T + (-6.06 - 3.5i)T^{2}
11 1+(2.79+0.748i)T+(9.52+5.5i)T2 1 + (2.79 + 0.748i)T + (9.52 + 5.5i)T^{2}
17 1+6.33T+17T2 1 + 6.33T + 17T^{2}
19 1+(−0.0431+0.0431i)T−19iT2 1 + (-0.0431 + 0.0431i)T - 19iT^{2}
23 1+(1.17−2.02i)T+(−11.5−19.9i)T2 1 + (1.17 - 2.02i)T + (-11.5 - 19.9i)T^{2}
29 1+(−8.31+4.80i)T+(14.5−25.1i)T2 1 + (-8.31 + 4.80i)T + (14.5 - 25.1i)T^{2}
31 1+(−0.602−2.24i)T+(−26.8+15.5i)T2 1 + (-0.602 - 2.24i)T + (-26.8 + 15.5i)T^{2}
37 1+(3.87+3.87i)T+37iT2 1 + (3.87 + 3.87i)T + 37iT^{2}
41 1+(−2.06+0.554i)T+(35.5−20.5i)T2 1 + (-2.06 + 0.554i)T + (35.5 - 20.5i)T^{2}
43 1+(−7.40+4.27i)T+(21.5−37.2i)T2 1 + (-7.40 + 4.27i)T + (21.5 - 37.2i)T^{2}
47 1+(−4.53−1.21i)T+(40.7+23.5i)T2 1 + (-4.53 - 1.21i)T + (40.7 + 23.5i)T^{2}
53 1−3.15iT−53T2 1 - 3.15iT - 53T^{2}
59 1+(0.734+2.74i)T+(−51.0+29.5i)T2 1 + (0.734 + 2.74i)T + (-51.0 + 29.5i)T^{2}
61 1+(4.45+7.71i)T+(−30.5+52.8i)T2 1 + (4.45 + 7.71i)T + (-30.5 + 52.8i)T^{2}
67 1+(3.23+12.0i)T+(−58.0+33.5i)T2 1 + (3.23 + 12.0i)T + (-58.0 + 33.5i)T^{2}
71 1+(−6.13−6.13i)T+71iT2 1 + (-6.13 - 6.13i)T + 71iT^{2}
73 1+(−0.333−0.333i)T+73iT2 1 + (-0.333 - 0.333i)T + 73iT^{2}
79 1+(−6.64−11.5i)T+(−39.5+68.4i)T2 1 + (-6.64 - 11.5i)T + (-39.5 + 68.4i)T^{2}
83 1+(1.46−5.45i)T+(−71.8−41.5i)T2 1 + (1.46 - 5.45i)T + (-71.8 - 41.5i)T^{2}
89 1+(4.52−4.52i)T−89iT2 1 + (4.52 - 4.52i)T - 89iT^{2}
97 1+(13.7+3.67i)T+(84.0+48.5i)T2 1 + (13.7 + 3.67i)T + (84.0 + 48.5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−13.60119323382196631620073561049, −12.24792527896517126361533063146, −11.03858870435070693989393185587, −10.69339395829467675304170555033, −9.413933662321959415649752351178, −8.075287219526762026085932819966, −7.05998201893677852076029425552, −4.52134247818335848875267024127, −3.81354764567519320577697332492, −2.38308041231881869072133355006, 2.53669219358594337073738208133, 4.57899159473941756463408613194, 5.99151765062679406972821408264, 7.18345886548515813700559696308, 8.134486229160459867053814555422, 8.705978717258750164544339416611, 10.56062851032966036241755761176, 11.90497766816820817117084221697, 12.89013277243776852243986254054, 13.80216895876508590501288572582

Graph of the ZZ-function along the critical line