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Properties

Label 2-612-204.203-c1-0-20
Degree 22
Conductor 612612
Sign 0.403+0.915i0.403 + 0.915i
Analytic cond. 4.886844.88684
Root an. cond. 2.210622.21062
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  − 1.41i·2-s − 2.00·4-s + 4.24·5-s + 2.82i·8-s − 6i·10-s + 4·13-s + 4.00·16-s + (−2.12 + 3.53i)17-s − 8.48·20-s + 12.9·25-s − 5.65i·26-s + 4.24·29-s − 5.65i·32-s + (5.00 + 3i)34-s − 12i·37-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s + 1.89·5-s + 1.00i·8-s − 1.89i·10-s + 1.10·13-s + 1.00·16-s + (−0.514 + 0.857i)17-s − 1.89·20-s + 2.59·25-s − 1.10i·26-s + 0.787·29-s − 1.00i·32-s + (0.857 + 0.514i)34-s − 1.97i·37-s + ⋯

Functional equation

Λ(s)=(612s/2 ΓC(s) L(s)=((0.403+0.915i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.403 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(612s/2 ΓC(s+1/2) L(s)=((0.403+0.915i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 612 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.403 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 612612    =    22⋅32⋅172^{2} \cdot 3^{2} \cdot 17
Sign: 0.403+0.915i0.403 + 0.915i
Analytic conductor: 4.886844.88684
Root analytic conductor: 2.210622.21062
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ612(611,⋅)\chi_{612} (611, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 612, ( :1/2), 0.403+0.915i)(2,\ 612,\ (\ :1/2),\ 0.403 + 0.915i)

Particular Values

L(1)L(1) ≈\approx 1.57413−1.02671i1.57413 - 1.02671i
L(12)L(\frac12) ≈\approx 1.57413−1.02671i1.57413 - 1.02671i
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+1.41iT 1 + 1.41iT
3 1 1
17 1+(2.12−3.53i)T 1 + (2.12 - 3.53i)T
good5 1−4.24T+5T2 1 - 4.24T + 5T^{2}
7 1+7T2 1 + 7T^{2}
11 1−11T2 1 - 11T^{2}
13 1−4T+13T2 1 - 4T + 13T^{2}
19 1−19T2 1 - 19T^{2}
23 1−23T2 1 - 23T^{2}
29 1−4.24T+29T2 1 - 4.24T + 29T^{2}
31 1+31T2 1 + 31T^{2}
37 1+12iT−37T2 1 + 12iT - 37T^{2}
41 1+12.7T+41T2 1 + 12.7T + 41T^{2}
43 1−43T2 1 - 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1−7.07iT−53T2 1 - 7.07iT - 53T^{2}
59 1+59T2 1 + 59T^{2}
61 1+12iT−61T2 1 + 12iT - 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1−71T2 1 - 71T^{2}
73 1+6iT−73T2 1 + 6iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+83T2 1 + 83T^{2}
89 1−18.3iT−89T2 1 - 18.3iT - 89T^{2}
97 1−18iT−97T2 1 - 18iT - 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

−10.57800020521757063207459569445, −9.711449739754592838307808238067, −9.019792580697769266187733693614, −8.293362797966283743041912031050, −6.56012848394888563171478145990, −5.82406983635837267046016990018, −4.90448517018594930684171028443, −3.57136937362775505058143052958, −2.29737296873673019062319256402, −1.41782464593145812618402367898, 1.43465017352053930939310262577, 3.05861174260679874024368265805, 4.70269060891022292300479711789, 5.47309714159647814068770226520, 6.39578760802427181320237802216, 6.82735631565073382983028544291, 8.351374099554834379360659646890, 8.933911177544503426285143134084, 9.856707297964413275165864859537, 10.33036184573592275883695698322

Graph of the ZZ-function along the critical line