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Properties

Label 2-152-152.107-c1-0-7
Degree 22
Conductor 152152
Sign 0.402−0.915i0.402 - 0.915i
Analytic cond. 1.213721.21372
Root an. cond. 1.101691.10169
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  + (1.14 + 0.834i)2-s + (−1.05 + 0.606i)3-s + (0.606 + 1.90i)4-s + (2.45 − 1.41i)5-s + (−1.70 − 0.184i)6-s + 0.450i·7-s + (−0.897 + 2.68i)8-s + (−0.763 + 1.32i)9-s + (3.97 + 0.429i)10-s − 2.15·11-s + (−1.79 − 1.63i)12-s + (1.86 − 3.22i)13-s + (−0.376 + 0.514i)14-s + (−1.71 + 2.97i)15-s + (−3.26 + 2.31i)16-s + (−0.716 − 1.24i)17-s + ⋯
L(s)  = 1  + (0.807 + 0.590i)2-s + (−0.606 + 0.350i)3-s + (0.303 + 0.952i)4-s + (1.09 − 0.632i)5-s + (−0.696 − 0.0752i)6-s + 0.170i·7-s + (−0.317 + 0.948i)8-s + (−0.254 + 0.440i)9-s + (1.25 + 0.135i)10-s − 0.651·11-s + (−0.517 − 0.471i)12-s + (0.516 − 0.895i)13-s + (−0.100 + 0.137i)14-s + (−0.443 + 0.767i)15-s + (−0.815 + 0.578i)16-s + (−0.173 − 0.301i)17-s + ⋯

Functional equation

Λ(s)=(152s/2 ΓC(s) L(s)=((0.402−0.915i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(152s/2 ΓC(s+1/2) L(s)=((0.402−0.915i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 152152    =    23⋅192^{3} \cdot 19
Sign: 0.402−0.915i0.402 - 0.915i
Analytic conductor: 1.213721.21372
Root analytic conductor: 1.101691.10169
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ152(107,⋅)\chi_{152} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 152, ( :1/2), 0.402−0.915i)(2,\ 152,\ (\ :1/2),\ 0.402 - 0.915i)

Particular Values

L(1)L(1) ≈\approx 1.30672+0.853290i1.30672 + 0.853290i
L(12)L(\frac12) ≈\approx 1.30672+0.853290i1.30672 + 0.853290i
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)
bad2 1+(−1.14−0.834i)T 1 + (-1.14 - 0.834i)T
19 1+(0.0305+4.35i)T 1 + (0.0305 + 4.35i)T
good3 1+(1.05−0.606i)T+(1.5−2.59i)T2 1 + (1.05 - 0.606i)T + (1.5 - 2.59i)T^{2}
5 1+(−2.45+1.41i)T+(2.5−4.33i)T2 1 + (-2.45 + 1.41i)T + (2.5 - 4.33i)T^{2}
7 1−0.450iT−7T2 1 - 0.450iT - 7T^{2}
11 1+2.15T+11T2 1 + 2.15T + 11T^{2}
13 1+(−1.86+3.22i)T+(−6.5−11.2i)T2 1 + (-1.86 + 3.22i)T + (-6.5 - 11.2i)T^{2}
17 1+(0.716+1.24i)T+(−8.5+14.7i)T2 1 + (0.716 + 1.24i)T + (-8.5 + 14.7i)T^{2}
23 1+(−1.12−0.652i)T+(11.5+19.9i)T2 1 + (-1.12 - 0.652i)T + (11.5 + 19.9i)T^{2}
29 1+(−4.22+7.32i)T+(−14.5−25.1i)T2 1 + (-4.22 + 7.32i)T + (-14.5 - 25.1i)T^{2}
31 1−0.497T+31T2 1 - 0.497T + 31T^{2}
37 1+6.72T+37T2 1 + 6.72T + 37T^{2}
41 1+(7.30−4.21i)T+(20.5−35.5i)T2 1 + (7.30 - 4.21i)T + (20.5 - 35.5i)T^{2}
43 1+(−2.90−5.02i)T+(−21.5+37.2i)T2 1 + (-2.90 - 5.02i)T + (-21.5 + 37.2i)T^{2}
47 1+(−0.567−0.327i)T+(23.5+40.7i)T2 1 + (-0.567 - 0.327i)T + (23.5 + 40.7i)T^{2}
53 1+(3.86−6.69i)T+(−26.5−45.8i)T2 1 + (3.86 - 6.69i)T + (-26.5 - 45.8i)T^{2}
59 1+(12.1−6.99i)T+(29.5−51.0i)T2 1 + (12.1 - 6.99i)T + (29.5 - 51.0i)T^{2}
61 1+(−2.13−1.23i)T+(30.5+52.8i)T2 1 + (-2.13 - 1.23i)T + (30.5 + 52.8i)T^{2}
67 1+(−2.16−1.25i)T+(33.5+58.0i)T2 1 + (-2.16 - 1.25i)T + (33.5 + 58.0i)T^{2}
71 1+(8.35+14.4i)T+(−35.5+61.4i)T2 1 + (8.35 + 14.4i)T + (-35.5 + 61.4i)T^{2}
73 1+(−8.25−14.2i)T+(−36.5+63.2i)T2 1 + (-8.25 - 14.2i)T + (-36.5 + 63.2i)T^{2}
79 1+(−4.05−7.02i)T+(−39.5+68.4i)T2 1 + (-4.05 - 7.02i)T + (-39.5 + 68.4i)T^{2}
83 1−11.8T+83T2 1 - 11.8T + 83T^{2}
89 1+(6.95+4.01i)T+(44.5+77.0i)T2 1 + (6.95 + 4.01i)T + (44.5 + 77.0i)T^{2}
97 1+(5.47−3.15i)T+(48.5−84.0i)T2 1 + (5.47 - 3.15i)T + (48.5 - 84.0i)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

−13.49874104898433333357144531158, −12.41823546719022349044123115496, −11.28137033306575078089345680898, −10.28909866422768409990854211743, −8.934228587946402497137413956243, −7.85689530462315369604837992685, −6.27012710975716018429223533342, −5.43850362650757765929005689913, −4.73289293067868230197829541798, −2.67158364991551557083930519531, 1.81839705723372484590095866751, 3.43666919629166138341634221442, 5.20621488747264355135106664206, 6.19379506101887001518413345674, 6.86694881440744323807171955919, 8.970317043293839796111238353376, 10.24924475797263584851759189922, 10.80222501285068231088609387581, 11.94075362968840413290431473904, 12.73168221751295288708139458914

Graph of the ZZ-function along the critical line