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Properties

Label 2-70-35.27-c1-0-1
Degree 22
Conductor 7070
Sign 0.709−0.704i0.709 - 0.704i
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  + (0.707 + 0.707i)2-s + (0.541 + 0.541i)3-s + 1.00i·4-s + (−2.23 + 0.158i)5-s + 0.765i·6-s + (1.55 − 2.14i)7-s + (−0.707 + 0.707i)8-s − 2.41i·9-s + (−1.68 − 1.46i)10-s − 2.82·11-s + (−0.541 + 0.541i)12-s + (2.83 + 2.83i)13-s + (2.61 − 0.414i)14-s + (−1.29 − 1.12i)15-s − 1.00·16-s + (1.53 − 1.53i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.312 + 0.312i)3-s + 0.500i·4-s + (−0.997 + 0.0708i)5-s + 0.312i·6-s + (0.587 − 0.809i)7-s + (−0.250 + 0.250i)8-s − 0.804i·9-s + (−0.534 − 0.463i)10-s − 0.852·11-s + (−0.156 + 0.156i)12-s + (0.786 + 0.786i)13-s + (0.698 − 0.110i)14-s + (−0.333 − 0.289i)15-s − 0.250·16-s + (0.371 − 0.371i)17-s + ⋯

Functional equation

Λ(s)=(70s/2 ΓC(s) L(s)=((0.709−0.704i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(70s/2 ΓC(s+1/2) L(s)=((0.709−0.704i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7070    =    2⋅5⋅72 \cdot 5 \cdot 7
Sign: 0.709−0.704i0.709 - 0.704i
Analytic conductor: 0.5589520.558952
Root analytic conductor: 0.7476310.747631
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ70(27,⋅)\chi_{70} (27, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 70, ( :1/2), 0.709−0.704i)(2,\ 70,\ (\ :1/2),\ 0.709 - 0.704i)

Particular Values

L(1)L(1) ≈\approx 1.03517+0.426560i1.03517 + 0.426560i
L(12)L(\frac12) ≈\approx 1.03517+0.426560i1.03517 + 0.426560i
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+(−0.707−0.707i)T 1 + (-0.707 - 0.707i)T
5 1+(2.23−0.158i)T 1 + (2.23 - 0.158i)T
7 1+(−1.55+2.14i)T 1 + (-1.55 + 2.14i)T
good3 1+(−0.541−0.541i)T+3iT2 1 + (-0.541 - 0.541i)T + 3iT^{2}
11 1+2.82T+11T2 1 + 2.82T + 11T^{2}
13 1+(−2.83−2.83i)T+13iT2 1 + (-2.83 - 2.83i)T + 13iT^{2}
17 1+(−1.53+1.53i)T−17iT2 1 + (-1.53 + 1.53i)T - 17iT^{2}
19 1+7.07T+19T2 1 + 7.07T + 19T^{2}
23 1+(2.41−2.41i)T−23iT2 1 + (2.41 - 2.41i)T - 23iT^{2}
29 1−4.82iT−29T2 1 - 4.82iT - 29T^{2}
31 1−3.69iT−31T2 1 - 3.69iT - 31T^{2}
37 1+(−5.41−5.41i)T+37iT2 1 + (-5.41 - 5.41i)T + 37iT^{2}
41 1+1.53iT−41T2 1 + 1.53iT - 41T^{2}
43 1+(−4+4i)T−43iT2 1 + (-4 + 4i)T - 43iT^{2}
47 1+(2.61−2.61i)T−47iT2 1 + (2.61 - 2.61i)T - 47iT^{2}
53 1+(−0.242+0.242i)T−53iT2 1 + (-0.242 + 0.242i)T - 53iT^{2}
59 1−3.82T+59T2 1 - 3.82T + 59T^{2}
61 1+10.3iT−61T2 1 + 10.3iT - 61T^{2}
67 1+(6.48+6.48i)T+67iT2 1 + (6.48 + 6.48i)T + 67iT^{2}
71 1+3.41T+71T2 1 + 3.41T + 71T^{2}
73 1+(−4.77−4.77i)T+73iT2 1 + (-4.77 - 4.77i)T + 73iT^{2}
79 1+9.07iT−79T2 1 + 9.07iT - 79T^{2}
83 1+(5.45+5.45i)T+83iT2 1 + (5.45 + 5.45i)T + 83iT^{2}
89 1−16.9T+89T2 1 - 16.9T + 89T^{2}
97 1+(11.0−11.0i)T−97iT2 1 + (11.0 - 11.0i)T - 97iT^{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

−14.86495883359545340948585141707, −14.04186313654186145371724770179, −12.78911009055174607590716488114, −11.61222589253796268710097186622, −10.55491785123610102053552139526, −8.788634649514958751010422261998, −7.79600326359745513209330029133, −6.57503830327895080777047985474, −4.61186103028457701914632051657, −3.58589607845968787871267683122, 2.49398922054417643061211859424, 4.34714517064839696655961222057, 5.80269736865371601942855919682, 7.85927429299248757757125324790, 8.487602544523923664017689457570, 10.51185031710145582993958243683, 11.29851779630329402381334620667, 12.55333840121385503185430017922, 13.21628724775994049998882943255, 14.65623325463360571351479656474

Graph of the ZZ-function along the critical line