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Properties

Label 2-700-28.19-c1-0-1
Degree 22
Conductor 700700
Sign 0.839−0.542i0.839 - 0.542i
Analytic cond. 5.589525.58952
Root an. cond. 2.364212.36421
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  + (−0.285 − 1.38i)2-s + (−1.29 − 2.24i)3-s + (−1.83 + 0.791i)4-s + (−2.74 + 2.44i)6-s + (0.603 − 2.57i)7-s + (1.62 + 2.31i)8-s + (−1.87 + 3.23i)9-s + (−3.12 + 1.80i)11-s + (4.16 + 3.10i)12-s + 0.818i·13-s + (−3.74 − 0.100i)14-s + (2.74 − 2.90i)16-s + (−6.40 + 3.69i)17-s + (5.02 + 1.66i)18-s + (−1.65 + 2.86i)19-s + ⋯
L(s)  = 1  + (−0.202 − 0.979i)2-s + (−0.749 − 1.29i)3-s + (−0.918 + 0.395i)4-s + (−1.11 + 0.996i)6-s + (0.228 − 0.973i)7-s + (0.573 + 0.819i)8-s + (−0.623 + 1.07i)9-s + (−0.940 + 0.543i)11-s + (1.20 + 0.895i)12-s + 0.226i·13-s + (−0.999 − 0.0267i)14-s + (0.686 − 0.727i)16-s + (−1.55 + 0.896i)17-s + (1.18 + 0.392i)18-s + (−0.379 + 0.656i)19-s + ⋯

Functional equation

Λ(s)=(700s/2 ΓC(s) L(s)=((0.839−0.542i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.542i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(700s/2 ΓC(s+1/2) L(s)=((0.839−0.542i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 - 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 700700    =    22⋅52⋅72^{2} \cdot 5^{2} \cdot 7
Sign: 0.839−0.542i0.839 - 0.542i
Analytic conductor: 5.589525.58952
Root analytic conductor: 2.364212.36421
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ700(551,⋅)\chi_{700} (551, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 700, ( :1/2), 0.839−0.542i)(2,\ 700,\ (\ :1/2),\ 0.839 - 0.542i)

Particular Values

L(1)L(1) ≈\approx 0.0782375+0.0230730i0.0782375 + 0.0230730i
L(12)L(\frac12) ≈\approx 0.0782375+0.0230730i0.0782375 + 0.0230730i
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+(0.285+1.38i)T 1 + (0.285 + 1.38i)T
5 1 1
7 1+(−0.603+2.57i)T 1 + (-0.603 + 2.57i)T
good3 1+(1.29+2.24i)T+(−1.5+2.59i)T2 1 + (1.29 + 2.24i)T + (-1.5 + 2.59i)T^{2}
11 1+(3.12−1.80i)T+(5.5−9.52i)T2 1 + (3.12 - 1.80i)T + (5.5 - 9.52i)T^{2}
13 1−0.818iT−13T2 1 - 0.818iT - 13T^{2}
17 1+(6.40−3.69i)T+(8.5−14.7i)T2 1 + (6.40 - 3.69i)T + (8.5 - 14.7i)T^{2}
19 1+(1.65−2.86i)T+(−9.5−16.4i)T2 1 + (1.65 - 2.86i)T + (-9.5 - 16.4i)T^{2}
23 1+(−2.19−1.26i)T+(11.5+19.9i)T2 1 + (-2.19 - 1.26i)T + (11.5 + 19.9i)T^{2}
29 1−2.04T+29T2 1 - 2.04T + 29T^{2}
31 1+(−0.955−1.65i)T+(−15.5+26.8i)T2 1 + (-0.955 - 1.65i)T + (-15.5 + 26.8i)T^{2}
37 1+(−3.58+6.20i)T+(−18.5−32.0i)T2 1 + (-3.58 + 6.20i)T + (-18.5 - 32.0i)T^{2}
41 1−2.65iT−41T2 1 - 2.65iT - 41T^{2}
43 1+2.39iT−43T2 1 + 2.39iT - 43T^{2}
47 1+(−0.667+1.15i)T+(−23.5−40.7i)T2 1 + (-0.667 + 1.15i)T + (-23.5 - 40.7i)T^{2}
53 1+(0.905+1.56i)T+(−26.5+45.8i)T2 1 + (0.905 + 1.56i)T + (-26.5 + 45.8i)T^{2}
59 1+(−0.955−1.65i)T+(−29.5+51.0i)T2 1 + (-0.955 - 1.65i)T + (-29.5 + 51.0i)T^{2}
61 1+(8.46+4.88i)T+(30.5+52.8i)T2 1 + (8.46 + 4.88i)T + (30.5 + 52.8i)T^{2}
67 1+(8.02−4.63i)T+(33.5−58.0i)T2 1 + (8.02 - 4.63i)T + (33.5 - 58.0i)T^{2}
71 1−1.38iT−71T2 1 - 1.38iT - 71T^{2}
73 1+(6.40−3.69i)T+(36.5−63.2i)T2 1 + (6.40 - 3.69i)T + (36.5 - 63.2i)T^{2}
79 1+(−6.70−3.87i)T+(39.5+68.4i)T2 1 + (-6.70 - 3.87i)T + (39.5 + 68.4i)T^{2}
83 1+10.4T+83T2 1 + 10.4T + 83T^{2}
89 1+(−9.19−5.30i)T+(44.5+77.0i)T2 1 + (-9.19 - 5.30i)T + (44.5 + 77.0i)T^{2}
97 1+7.32iT−97T2 1 + 7.32iT - 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

−10.78533865934138978758871078987, −10.02030265666431763261175748987, −8.740834115982041560187400129272, −7.84353711955032593741451426772, −7.18143657517573119109013869672, −6.17887440451359413579161697720, −4.93703815883524410958874646378, −4.00438915953378249908170777007, −2.34884787402766119329566969672, −1.39665161026020432350178332210, 0.05216694723263177385890307820, 2.83153165793376765578690330858, 4.50187091487278627424846641118, 4.92545968432631411043060869969, 5.78717150538435654247947067448, 6.57354695378038961254524633951, 7.86578791857027914882602087724, 8.864412163142602567491151821995, 9.285218709643996299509484001421, 10.35062235677126706803270671185

Graph of the ZZ-function along the critical line