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Properties

Label 2-850-17.13-c1-0-17
Degree 22
Conductor 850850
Sign −0.122+0.992i-0.122 + 0.992i
Analytic cond. 6.787286.78728
Root an. cond. 2.605242.60524
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 + (0.292 + 0.292i)3-s − 4-s + (0.292 − 0.292i)6-s + (−1 + i)7-s + i·8-s − 2.82i·9-s + (−1.58 + 1.58i)11-s + (−0.292 − 0.292i)12-s + 3·13-s + (1 + i)14-s + 16-s + (3.53 − 2.12i)17-s − 2.82·18-s − 7.24i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.169 + 0.169i)3-s − 0.5·4-s + (0.119 − 0.119i)6-s + (−0.377 + 0.377i)7-s + 0.353i·8-s − 0.942i·9-s + (−0.478 + 0.478i)11-s + (−0.0845 − 0.0845i)12-s + 0.832·13-s + (0.267 + 0.267i)14-s + 0.250·16-s + (0.857 − 0.514i)17-s − 0.666·18-s − 1.66i·19-s + ⋯

Functional equation

Λ(s)=(850s/2 ΓC(s) L(s)=((−0.122+0.992i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(850s/2 ΓC(s+1/2) L(s)=((−0.122+0.992i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 850850    =    2⋅52⋅172 \cdot 5^{2} \cdot 17
Sign: −0.122+0.992i-0.122 + 0.992i
Analytic conductor: 6.787286.78728
Root analytic conductor: 2.605242.60524
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ850(251,⋅)\chi_{850} (251, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 850, ( :1/2), −0.122+0.992i)(2,\ 850,\ (\ :1/2),\ -0.122 + 0.992i)

Particular Values

L(1)L(1) ≈\approx 0.922327−1.04283i0.922327 - 1.04283i
L(12)L(\frac12) ≈\approx 0.922327−1.04283i0.922327 - 1.04283i
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 1
17 1+(−3.53+2.12i)T 1 + (-3.53 + 2.12i)T
good3 1+(−0.292−0.292i)T+3iT2 1 + (-0.292 - 0.292i)T + 3iT^{2}
7 1+(1−i)T−7iT2 1 + (1 - i)T - 7iT^{2}
11 1+(1.58−1.58i)T−11iT2 1 + (1.58 - 1.58i)T - 11iT^{2}
13 1−3T+13T2 1 - 3T + 13T^{2}
19 1+7.24iT−19T2 1 + 7.24iT - 19T^{2}
23 1+(−2.82+2.82i)T−23iT2 1 + (-2.82 + 2.82i)T - 23iT^{2}
29 1+(0.707+0.707i)T+29iT2 1 + (0.707 + 0.707i)T + 29iT^{2}
31 1+(5.36+5.36i)T+31iT2 1 + (5.36 + 5.36i)T + 31iT^{2}
37 1+(−5.24−5.24i)T+37iT2 1 + (-5.24 - 5.24i)T + 37iT^{2}
41 1+(−4.41+4.41i)T−41iT2 1 + (-4.41 + 4.41i)T - 41iT^{2}
43 1+3.75iT−43T2 1 + 3.75iT - 43T^{2}
47 1−1.58T+47T2 1 - 1.58T + 47T^{2}
53 1−3iT−53T2 1 - 3iT - 53T^{2}
59 1+12.8iT−59T2 1 + 12.8iT - 59T^{2}
61 1+(−6.12+6.12i)T−61iT2 1 + (-6.12 + 6.12i)T - 61iT^{2}
67 1+14.4T+67T2 1 + 14.4T + 67T^{2}
71 1+(−3.70−3.70i)T+71iT2 1 + (-3.70 - 3.70i)T + 71iT^{2}
73 1+(−8.36−8.36i)T+73iT2 1 + (-8.36 - 8.36i)T + 73iT^{2}
79 1+(−0.242+0.242i)T−79iT2 1 + (-0.242 + 0.242i)T - 79iT^{2}
83 1+4.24iT−83T2 1 + 4.24iT - 83T^{2}
89 1+11.4T+89T2 1 + 11.4T + 89T^{2}
97 1+(−0.121−0.121i)T+97iT2 1 + (-0.121 - 0.121i)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

−9.792928641929643171930948952500, −9.316035858628484906174804627578, −8.594587430417202139576266044506, −7.44613429980914007088852648104, −6.44291292838065701549130433527, −5.42544197220060500323131489109, −4.36826912728141451748654689284, −3.31605220674106955059622117537, −2.47740990261862954494754564337, −0.73749579370959235806260053374, 1.42945286990051811107075447952, 3.16656000560235816804874695653, 4.08068577399684227193699904427, 5.45141151500490773865980505005, 5.92607948608329026762596753526, 7.14138543116041787564182575874, 7.86983604127261168191530144105, 8.433605096009157564431371012317, 9.478969152381042346448325714617, 10.44400455177865267159835543640

Graph of the ZZ-function along the critical line