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Properties

Label 2-145-29.28-c1-0-3
Degree 22
Conductor 145145
Sign 0.0556−0.998i0.0556 - 0.998i
Analytic cond. 1.157831.15783
Root an. cond. 1.076021.07602
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  + 1.30i·2-s + 0.537i·3-s + 0.299·4-s + 5-s − 0.700·6-s − 1.29·7-s + 2.99i·8-s + 2.71·9-s + 1.30i·10-s − 0.537i·11-s + 0.160i·12-s − 5.71·13-s − 1.69i·14-s + 0.537i·15-s − 3.31·16-s − 6.91i·17-s + ⋯
L(s)  = 1  + 0.922i·2-s + 0.310i·3-s + 0.149·4-s + 0.447·5-s − 0.285·6-s − 0.491·7-s + 1.06i·8-s + 0.903·9-s + 0.412i·10-s − 0.161i·11-s + 0.0464i·12-s − 1.58·13-s − 0.452i·14-s + 0.138i·15-s − 0.827·16-s − 1.67i·17-s + ⋯

Functional equation

Λ(s)=(145s/2 ΓC(s) L(s)=((0.0556−0.998i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0556 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(145s/2 ΓC(s+1/2) L(s)=((0.0556−0.998i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0556 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 145145    =    5⋅295 \cdot 29
Sign: 0.0556−0.998i0.0556 - 0.998i
Analytic conductor: 1.157831.15783
Root analytic conductor: 1.076021.07602
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ145(86,⋅)\chi_{145} (86, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 145, ( :1/2), 0.0556−0.998i)(2,\ 145,\ (\ :1/2),\ 0.0556 - 0.998i)

Particular Values

L(1)L(1) ≈\approx 0.916560+0.866900i0.916560 + 0.866900i
L(12)L(\frac12) ≈\approx 0.916560+0.866900i0.916560 + 0.866900i
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)
bad5 1−T 1 - T
29 1+(−0.299+5.37i)T 1 + (-0.299 + 5.37i)T
good2 1−1.30iT−2T2 1 - 1.30iT - 2T^{2}
3 1−0.537iT−3T2 1 - 0.537iT - 3T^{2}
7 1+1.29T+7T2 1 + 1.29T + 7T^{2}
11 1+0.537iT−11T2 1 + 0.537iT - 11T^{2}
13 1+5.71T+13T2 1 + 5.71T + 13T^{2}
17 1+6.91iT−17T2 1 + 6.91iT - 17T^{2}
19 1−4.30iT−19T2 1 - 4.30iT - 19T^{2}
23 1−5.01T+23T2 1 - 5.01T + 23T^{2}
31 1+8.44iT−31T2 1 + 8.44iT - 31T^{2}
37 1+7.98iT−37T2 1 + 7.98iT - 37T^{2}
41 1+0.698iT−41T2 1 + 0.698iT - 41T^{2}
43 1−10.2iT−43T2 1 - 10.2iT - 43T^{2}
47 1+0.537iT−47T2 1 + 0.537iT - 47T^{2}
53 1−7.11T+53T2 1 - 7.11T + 53T^{2}
59 1+7.71T+59T2 1 + 7.71T + 59T^{2}
61 1−5.91iT−61T2 1 - 5.91iT - 61T^{2}
67 1+9.01T+67T2 1 + 9.01T + 67T^{2}
71 1+4.28T+71T2 1 + 4.28T + 71T^{2}
73 1−6.21iT−73T2 1 - 6.21iT - 73T^{2}
79 1−10.2iT−79T2 1 - 10.2iT - 79T^{2}
83 1+2.70T+83T2 1 + 2.70T + 83T^{2}
89 1−9.67iT−89T2 1 - 9.67iT - 89T^{2}
97 1+2.07iT−97T2 1 + 2.07iT - 97T^{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

−13.52009534586050623017846878307, −12.41018957565350608735111252967, −11.31273117567351878626936998791, −9.960339939109492255377409886476, −9.354268791870579598614692943461, −7.66425740271153784813463286524, −7.02240022142731459812814913238, −5.77454810245506480204364415517, −4.65466071700818048037903256614, −2.57949465807300233005271469073, 1.69718507159421331233031470623, 3.10748322824270907517974505288, 4.76583666621125208548521017164, 6.58236105731666068803614645463, 7.25194686903495725995506576691, 9.035105806642557997769587241157, 10.12079407181254602969244465614, 10.61572978638732808332179461419, 12.08343053442731732988021798043, 12.67462688514460083704161215431

Graph of the ZZ-function along the critical line