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Properties

Label 2-850-17.13-c1-0-7
Degree 22
Conductor 850850
Sign 0.122−0.992i0.122 - 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 + (1.70 + 1.70i)3-s − 4-s + (1.70 − 1.70i)6-s + (−1 + i)7-s + i·8-s + 2.82i·9-s + (−4.41 + 4.41i)11-s + (−1.70 − 1.70i)12-s + 3·13-s + (1 + i)14-s + 16-s + (−3.53 + 2.12i)17-s + 2.82·18-s + 1.24i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.985 + 0.985i)3-s − 0.5·4-s + (0.696 − 0.696i)6-s + (−0.377 + 0.377i)7-s + 0.353i·8-s + 0.942i·9-s + (−1.33 + 1.33i)11-s + (−0.492 − 0.492i)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 + 0.285i·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: 0.122−0.992i0.122 - 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 1.16829+1.03329i1.16829 + 1.03329i
L(12)L(\frac12) ≈\approx 1.16829+1.03329i1.16829 + 1.03329i
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+(−1.70−1.70i)T+3iT2 1 + (-1.70 - 1.70i)T + 3iT^{2}
7 1+(1−i)T−7iT2 1 + (1 - i)T - 7iT^{2}
11 1+(4.41−4.41i)T−11iT2 1 + (4.41 - 4.41i)T - 11iT^{2}
13 1−3T+13T2 1 - 3T + 13T^{2}
19 1−1.24iT−19T2 1 - 1.24iT - 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+(−7.36−7.36i)T+31iT2 1 + (-7.36 - 7.36i)T + 31iT^{2}
37 1+(3.24+3.24i)T+37iT2 1 + (3.24 + 3.24i)T + 37iT^{2}
41 1+(−1.58+1.58i)T−41iT2 1 + (-1.58 + 1.58i)T - 41iT^{2}
43 1+12.2iT−43T2 1 + 12.2iT - 43T^{2}
47 1−4.41T+47T2 1 - 4.41T + 47T^{2}
53 1−3iT−53T2 1 - 3iT - 53T^{2}
59 1−6.89iT−59T2 1 - 6.89iT - 59T^{2}
61 1+(−1.87+1.87i)T−61iT2 1 + (-1.87 + 1.87i)T - 61iT^{2}
67 1−2.48T+67T2 1 - 2.48T + 67T^{2}
71 1+(−2.29−2.29i)T+71iT2 1 + (-2.29 - 2.29i)T + 71iT^{2}
73 1+(4.36+4.36i)T+73iT2 1 + (4.36 + 4.36i)T + 73iT^{2}
79 1+(8.24−8.24i)T−79iT2 1 + (8.24 - 8.24i)T - 79iT^{2}
83 1−4.24iT−83T2 1 - 4.24iT - 83T^{2}
89 1−5.48T+89T2 1 - 5.48T + 89T^{2}
97 1+(4.12+4.12i)T+97iT2 1 + (4.12 + 4.12i)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

−10.32385633440884296043644101620, −9.674708386917633464537013254231, −8.802634427448273357752119410690, −8.299119314179656622759427767009, −7.14421767467650115528285186245, −5.73700370047887424424130565433, −4.67167820542052246345106106572, −3.90227813510474451603400544847, −2.91116516467249981356759512570, −2.03227463506447347816024187161, 0.65710634214172662455328054846, 2.46200061442239177351263155582, 3.33267270459464070028522390797, 4.65227444461460451360459633276, 5.97929583843138150803825606952, 6.58999761359573228457737170864, 7.59002070891298547810864378977, 8.256847447933813019895573484881, 8.656077595071778500645043837184, 9.773673771962375245665599334870

Graph of the ZZ-function along the critical line