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Properties

Label 2-370-185.117-c1-0-8
Degree 22
Conductor 370370
Sign 0.575−0.817i0.575 - 0.817i
Analytic cond. 2.954462.95446
Root an. cond. 1.718851.71885
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  + 2-s + (1 + i)3-s + 4-s + (−1 + 2i)5-s + (1 + i)6-s + (1 + i)7-s + 8-s − i·9-s + (−1 + 2i)10-s − 2i·11-s + (1 + i)12-s + 2·13-s + (1 + i)14-s + (−3 + i)15-s + 16-s + 4i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.577 + 0.577i)3-s + 0.5·4-s + (−0.447 + 0.894i)5-s + (0.408 + 0.408i)6-s + (0.377 + 0.377i)7-s + 0.353·8-s − 0.333i·9-s + (−0.316 + 0.632i)10-s − 0.603i·11-s + (0.288 + 0.288i)12-s + 0.554·13-s + (0.267 + 0.267i)14-s + (−0.774 + 0.258i)15-s + 0.250·16-s + 0.970i·17-s + ⋯

Functional equation

Λ(s)=(370s/2 ΓC(s) L(s)=((0.575−0.817i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(370s/2 ΓC(s+1/2) L(s)=((0.575−0.817i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 370370    =    2⋅5⋅372 \cdot 5 \cdot 37
Sign: 0.575−0.817i0.575 - 0.817i
Analytic conductor: 2.954462.95446
Root analytic conductor: 1.718851.71885
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ370(117,⋅)\chi_{370} (117, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 370, ( :1/2), 0.575−0.817i)(2,\ 370,\ (\ :1/2),\ 0.575 - 0.817i)

Particular Values

L(1)L(1) ≈\approx 2.05922+1.06930i2.05922 + 1.06930i
L(12)L(\frac12) ≈\approx 2.05922+1.06930i2.05922 + 1.06930i
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−T 1 - T
5 1+(1−2i)T 1 + (1 - 2i)T
37 1+(1+6i)T 1 + (1 + 6i)T
good3 1+(−1−i)T+3iT2 1 + (-1 - i)T + 3iT^{2}
7 1+(−1−i)T+7iT2 1 + (-1 - i)T + 7iT^{2}
11 1+2iT−11T2 1 + 2iT - 11T^{2}
13 1−2T+13T2 1 - 2T + 13T^{2}
17 1−4iT−17T2 1 - 4iT - 17T^{2}
19 1+(5−5i)T−19iT2 1 + (5 - 5i)T - 19iT^{2}
23 1+23T2 1 + 23T^{2}
29 1+(3+3i)T+29iT2 1 + (3 + 3i)T + 29iT^{2}
31 1+(−7+7i)T−31iT2 1 + (-7 + 7i)T - 31iT^{2}
41 1−41T2 1 - 41T^{2}
43 1−4T+43T2 1 - 4T + 43T^{2}
47 1+(7+7i)T+47iT2 1 + (7 + 7i)T + 47iT^{2}
53 1+(−1+i)T−53iT2 1 + (-1 + i)T - 53iT^{2}
59 1+(1−i)T−59iT2 1 + (1 - i)T - 59iT^{2}
61 1+(3−3i)T−61iT2 1 + (3 - 3i)T - 61iT^{2}
67 1+(−3+3i)T−67iT2 1 + (-3 + 3i)T - 67iT^{2}
71 1+8T+71T2 1 + 8T + 71T^{2}
73 1+(−9−9i)T+73iT2 1 + (-9 - 9i)T + 73iT^{2}
79 1+(1−i)T−79iT2 1 + (1 - i)T - 79iT^{2}
83 1+(5−5i)T−83iT2 1 + (5 - 5i)T - 83iT^{2}
89 1+(−5−5i)T+89iT2 1 + (-5 - 5i)T + 89iT^{2}
97 1−8iT−97T2 1 - 8iT - 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

−11.51681049479976715515495458736, −10.72214498184088405583760294808, −9.913561959008795010014553711726, −8.550678622946176948223109707713, −7.968092296445966825260904727649, −6.49397020060739909292475709936, −5.84108691251726407248319257508, −4.10728915493454226033796375498, −3.64159708286768808345882270281, −2.32661717346622627370675610005, 1.48399474842316005654978594715, 2.91408694033091515323513245375, 4.46225439661671786023374444253, 4.99367285908152138982104726730, 6.61342180482473715016656383816, 7.49965213761600358516025530418, 8.309662155208741088838408339798, 9.173798326193050005880679000962, 10.60314747515121830693942006683, 11.43854637309066998618799233508

Graph of the ZZ-function along the critical line