↑

Properties

Label 2-936-104.99-c1-0-44
Degree 22
Conductor 936936
Sign 0.957−0.289i0.957 - 0.289i
Analytic cond. 7.473997.47399
Root an. cond. 2.733862.73386
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s − 2.00·4-s + (2.82 − 2.82i)5-s + (1 + i)7-s − 2.82i·8-s + (4.00 + 4.00i)10-s + (−1.41 + 1.41i)11-s + (−2 − 3i)13-s + (−1.41 + 1.41i)14-s + 4.00·16-s + 2.82i·17-s + (1 + i)19-s + (−5.65 + 5.65i)20-s + (−2.00 − 2.00i)22-s + 8.48·23-s + ⋯
L(s)  = 1  + 0.999i·2-s − 1.00·4-s + (1.26 − 1.26i)5-s + (0.377 + 0.377i)7-s − 1.00i·8-s + (1.26 + 1.26i)10-s + (−0.426 + 0.426i)11-s + (−0.554 − 0.832i)13-s + (−0.377 + 0.377i)14-s + 1.00·16-s + 0.685i·17-s + (0.229 + 0.229i)19-s + (−1.26 + 1.26i)20-s + (−0.426 − 0.426i)22-s + 1.76·23-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((0.957−0.289i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s+1/2) L(s)=((0.957−0.289i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: 0.957−0.289i0.957 - 0.289i
Analytic conductor: 7.473997.47399
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ936(307,⋅)\chi_{936} (307, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :1/2), 0.957−0.289i)(2,\ 936,\ (\ :1/2),\ 0.957 - 0.289i)

Particular Values

L(1)L(1) ≈\approx 1.77050+0.262156i1.77050 + 0.262156i
L(12)L(\frac12) ≈\approx 1.77050+0.262156i1.77050 + 0.262156i
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 1−1.41iT 1 - 1.41iT
3 1 1
13 1+(2+3i)T 1 + (2 + 3i)T
good5 1+(−2.82+2.82i)T−5iT2 1 + (-2.82 + 2.82i)T - 5iT^{2}
7 1+(−1−i)T+7iT2 1 + (-1 - i)T + 7iT^{2}
11 1+(1.41−1.41i)T−11iT2 1 + (1.41 - 1.41i)T - 11iT^{2}
17 1−2.82iT−17T2 1 - 2.82iT - 17T^{2}
19 1+(−1−i)T+19iT2 1 + (-1 - i)T + 19iT^{2}
23 1−8.48T+23T2 1 - 8.48T + 23T^{2}
29 1+5.65iT−29T2 1 + 5.65iT - 29T^{2}
31 1+(−7+7i)T−31iT2 1 + (-7 + 7i)T - 31iT^{2}
37 1+(5+5i)T+37iT2 1 + (5 + 5i)T + 37iT^{2}
41 1+(−5.65−5.65i)T+41iT2 1 + (-5.65 - 5.65i)T + 41iT^{2}
43 1−43T2 1 - 43T^{2}
47 1+(−1.41−1.41i)T+47iT2 1 + (-1.41 - 1.41i)T + 47iT^{2}
53 1−2.82iT−53T2 1 - 2.82iT - 53T^{2}
59 1+(−7.07+7.07i)T−59iT2 1 + (-7.07 + 7.07i)T - 59iT^{2}
61 1−6iT−61T2 1 - 6iT - 61T^{2}
67 1+(5+5i)T+67iT2 1 + (5 + 5i)T + 67iT^{2}
71 1+(9.89−9.89i)T−71iT2 1 + (9.89 - 9.89i)T - 71iT^{2}
73 1+(−7+7i)T−73iT2 1 + (-7 + 7i)T - 73iT^{2}
79 1−6iT−79T2 1 - 6iT - 79T^{2}
83 1+(−9.89−9.89i)T+83iT2 1 + (-9.89 - 9.89i)T + 83iT^{2}
89 1+(5.65−5.65i)T−89iT2 1 + (5.65 - 5.65i)T - 89iT^{2}
97 1+(−7−7i)T+97iT2 1 + (-7 - 7i)T + 97iT^{2}
show more
show less
   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

−9.741516045703113180368052515879, −9.232557814808922384557791634634, −8.338447891430995512343439930492, −7.76331172524009456748299329849, −6.51168360143296946647637526215, −5.57609611317133313243438715147, −5.18649958556252205421469421587, −4.30622387659338216239599060642, −2.47145196150940810174359892889, −0.983196778336549892171486708245, 1.39910866146924081190762187500, 2.62707875287820081402961017309, 3.18179771749013310355984119111, 4.73945756716262267737653541427, 5.43136609808578778282734053478, 6.70072958320491420971095955299, 7.33279664106045445229271615226, 8.758204894563282903989929770427, 9.379467489961929225985018161623, 10.28063955721383013746549356219

Graph of the ZZ-function along the critical line