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Properties

Label 2-117-117.11-c1-0-2
Degree 22
Conductor 117117
Sign 0.450−0.892i0.450 - 0.892i
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  + (0.375 − 0.375i)2-s + (−0.0440 + 1.73i)3-s + 1.71i·4-s + (−2.85 − 0.764i)5-s + (0.633 + 0.667i)6-s + (3.85 + 1.03i)7-s + (1.39 + 1.39i)8-s + (−2.99 − 0.152i)9-s + (−1.35 + 0.784i)10-s + (1.41 + 1.41i)11-s + (−2.97 − 0.0757i)12-s + (−0.867 − 3.49i)13-s + (1.83 − 1.05i)14-s + (1.44 − 4.90i)15-s − 2.38·16-s + (2.38 − 4.12i)17-s + ⋯
L(s)  = 1  + (0.265 − 0.265i)2-s + (−0.0254 + 0.999i)3-s + 0.858i·4-s + (−1.27 − 0.341i)5-s + (0.258 + 0.272i)6-s + (1.45 + 0.390i)7-s + (0.493 + 0.493i)8-s + (−0.998 − 0.0508i)9-s + (−0.429 + 0.248i)10-s + (0.428 + 0.428i)11-s + (−0.858 − 0.0218i)12-s + (−0.240 − 0.970i)13-s + (0.490 − 0.283i)14-s + (0.374 − 1.26i)15-s − 0.596·16-s + (0.577 − 1.00i)17-s + ⋯

Functional equation

Λ(s)=(117s/2 ΓC(s) L(s)=((0.450−0.892i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(117s/2 ΓC(s+1/2) L(s)=((0.450−0.892i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 117117    =    32⋅133^{2} \cdot 13
Sign: 0.450−0.892i0.450 - 0.892i
Analytic conductor: 0.9342490.934249
Root analytic conductor: 0.9665650.966565
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ117(11,⋅)\chi_{117} (11, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 117, ( :1/2), 0.450−0.892i)(2,\ 117,\ (\ :1/2),\ 0.450 - 0.892i)

Particular Values

L(1)L(1) ≈\approx 0.939932+0.578534i0.939932 + 0.578534i
L(12)L(\frac12) ≈\approx 0.939932+0.578534i0.939932 + 0.578534i
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+(0.0440−1.73i)T 1 + (0.0440 - 1.73i)T
13 1+(0.867+3.49i)T 1 + (0.867 + 3.49i)T
good2 1+(−0.375+0.375i)T−2iT2 1 + (-0.375 + 0.375i)T - 2iT^{2}
5 1+(2.85+0.764i)T+(4.33+2.5i)T2 1 + (2.85 + 0.764i)T + (4.33 + 2.5i)T^{2}
7 1+(−3.85−1.03i)T+(6.06+3.5i)T2 1 + (-3.85 - 1.03i)T + (6.06 + 3.5i)T^{2}
11 1+(−1.41−1.41i)T+11iT2 1 + (-1.41 - 1.41i)T + 11iT^{2}
17 1+(−2.38+4.12i)T+(−8.5−14.7i)T2 1 + (-2.38 + 4.12i)T + (-8.5 - 14.7i)T^{2}
19 1+(−3.57+0.958i)T+(16.4−9.5i)T2 1 + (-3.57 + 0.958i)T + (16.4 - 9.5i)T^{2}
23 1+(0.469−0.812i)T+(−11.5−19.9i)T2 1 + (0.469 - 0.812i)T + (-11.5 - 19.9i)T^{2}
29 1+4.41iT−29T2 1 + 4.41iT - 29T^{2}
31 1+(−0.0175+0.0653i)T+(−26.8−15.5i)T2 1 + (-0.0175 + 0.0653i)T + (-26.8 - 15.5i)T^{2}
37 1+(0.00215+0.000576i)T+(32.0+18.5i)T2 1 + (0.00215 + 0.000576i)T + (32.0 + 18.5i)T^{2}
41 1+(−2.08−7.77i)T+(−35.5+20.5i)T2 1 + (-2.08 - 7.77i)T + (-35.5 + 20.5i)T^{2}
43 1+(8.64−4.99i)T+(21.5−37.2i)T2 1 + (8.64 - 4.99i)T + (21.5 - 37.2i)T^{2}
47 1+(−5.69+1.52i)T+(40.7−23.5i)T2 1 + (-5.69 + 1.52i)T + (40.7 - 23.5i)T^{2}
53 1−7.99iT−53T2 1 - 7.99iT - 53T^{2}
59 1+(3.08+3.08i)T+59iT2 1 + (3.08 + 3.08i)T + 59iT^{2}
61 1+(6.15+10.6i)T+(−30.5+52.8i)T2 1 + (6.15 + 10.6i)T + (-30.5 + 52.8i)T^{2}
67 1+(−1.05+0.282i)T+(58.0−33.5i)T2 1 + (-1.05 + 0.282i)T + (58.0 - 33.5i)T^{2}
71 1+(4.06+15.1i)T+(−61.4+35.5i)T2 1 + (4.06 + 15.1i)T + (-61.4 + 35.5i)T^{2}
73 1+(−0.854+0.854i)T−73iT2 1 + (-0.854 + 0.854i)T - 73iT^{2}
79 1+(0.501−0.868i)T+(−39.5−68.4i)T2 1 + (0.501 - 0.868i)T + (-39.5 - 68.4i)T^{2}
83 1+(−2.06−7.70i)T+(−71.8+41.5i)T2 1 + (-2.06 - 7.70i)T + (-71.8 + 41.5i)T^{2}
89 1+(−0.189+0.708i)T+(−77.0−44.5i)T2 1 + (-0.189 + 0.708i)T + (-77.0 - 44.5i)T^{2}
97 1+(−1.36+5.11i)T+(−84.0−48.5i)T2 1 + (-1.36 + 5.11i)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

−13.81533768439071521526580892970, −12.17335063934245424915782162336, −11.74642603103827370190573369958, −11.01078176965640963132008041378, −9.408279017897609053891446867420, −8.119230309631761807304332003585, −7.72781150062762627889927311392, −5.10733100054584930218062129094, −4.40140465889313103767997190962, −3.10177504672045394850138986062, 1.45878292505398273536679279012, 4.01352194994265877533215451198, 5.44057095585180127814355752953, 6.87700110979854852072995322693, 7.65492901892683715858986221782, 8.697511647547752058966101291362, 10.60581751924078597260850553884, 11.45891631460153480397516339162, 12.09121508498959308312894972671, 13.69632202817163985954759044000

Graph of the ZZ-function along the critical line