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Properties

Label 2-70-5.4-c1-0-1
Degree 22
Conductor 7070
Sign 0.994−0.100i0.994 - 0.100i
Analytic cond. 0.5589520.558952
Root an. cond. 0.7476310.747631
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  − i·2-s + 2.44i·3-s − 4-s + (2.22 − 0.224i)5-s + 2.44·6-s − i·7-s + i·8-s − 2.99·9-s + (−0.224 − 2.22i)10-s − 4.89·11-s − 2.44i·12-s − 4.44i·13-s − 14-s + (0.550 + 5.44i)15-s + 16-s + 2i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.41i·3-s − 0.5·4-s + (0.994 − 0.100i)5-s + 0.999·6-s − 0.377i·7-s + 0.353i·8-s − 0.999·9-s + (−0.0710 − 0.703i)10-s − 1.47·11-s − 0.707i·12-s − 1.23i·13-s − 0.267·14-s + (0.142 + 1.40i)15-s + 0.250·16-s + 0.485i·17-s + ⋯

Functional equation

Λ(s)=(70s/2 ΓC(s) L(s)=((0.994−0.100i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(70s/2 ΓC(s+1/2) L(s)=((0.994−0.100i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7070    =    2⋅5⋅72 \cdot 5 \cdot 7
Sign: 0.994−0.100i0.994 - 0.100i
Analytic conductor: 0.5589520.558952
Root analytic conductor: 0.7476310.747631
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ70(29,⋅)\chi_{70} (29, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 70, ( :1/2), 0.994−0.100i)(2,\ 70,\ (\ :1/2),\ 0.994 - 0.100i)

Particular Values

L(1)L(1) ≈\approx 0.942687+0.0474944i0.942687 + 0.0474944i
L(12)L(\frac12) ≈\approx 0.942687+0.0474944i0.942687 + 0.0474944i
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+iT 1 + iT
5 1+(−2.22+0.224i)T 1 + (-2.22 + 0.224i)T
7 1+iT 1 + iT
good3 1−2.44iT−3T2 1 - 2.44iT - 3T^{2}
11 1+4.89T+11T2 1 + 4.89T + 11T^{2}
13 1+4.44iT−13T2 1 + 4.44iT - 13T^{2}
17 1−2iT−17T2 1 - 2iT - 17T^{2}
19 1+1.55T+19T2 1 + 1.55T + 19T^{2}
23 1+2.89iT−23T2 1 + 2.89iT - 23T^{2}
29 1+6.89T+29T2 1 + 6.89T + 29T^{2}
31 1−8.89T+31T2 1 - 8.89T + 31T^{2}
37 1−2iT−37T2 1 - 2iT - 37T^{2}
41 1+1.10T+41T2 1 + 1.10T + 41T^{2}
43 1−0.898iT−43T2 1 - 0.898iT - 43T^{2}
47 1−8.89iT−47T2 1 - 8.89iT - 47T^{2}
53 1−10.8iT−53T2 1 - 10.8iT - 53T^{2}
59 1−1.55T+59T2 1 - 1.55T + 59T^{2}
61 1−3.55T+61T2 1 - 3.55T + 61T^{2}
67 1+8iT−67T2 1 + 8iT - 67T^{2}
71 1+1.10T+71T2 1 + 1.10T + 71T^{2}
73 1+2.89iT−73T2 1 + 2.89iT - 73T^{2}
79 1+6.89T+79T2 1 + 6.89T + 79T^{2}
83 1−2.44iT−83T2 1 - 2.44iT - 83T^{2}
89 1−10T+89T2 1 - 10T + 89T^{2}
97 1−15.7iT−97T2 1 - 15.7iT - 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

−14.80967854907105242526667714811, −13.50002272829615240033097414624, −12.69730178009787096880303935153, −10.73426123510157600035228413424, −10.43134582684158402341125059784, −9.522611812166983285514215208849, −8.163324242375110243183970953295, −5.69448156402581075408846634244, −4.61801799198574011265464084909, −2.90486681587310966934221475522, 2.20761612620777380177549525069, 5.29435469974098243385809056628, 6.44253967075169279419816373286, 7.41215864755701840767877718374, 8.648810880927592417393614859814, 9.972584520176235140764895497419, 11.66693394927099345644334706199, 13.01986328622609529127605007166, 13.45973863280713756712153726458, 14.44646370975234255480349180264

Graph of the ZZ-function along the critical line