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Properties

Label 2-936-39.20-c1-0-3
Degree 22
Conductor 936936
Sign 0.773−0.634i0.773 - 0.634i
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.661 + 0.661i)5-s + (−2.79 − 0.750i)7-s + (4.25 − 1.13i)11-s + (1.70 + 3.17i)13-s + (−0.870 − 1.50i)17-s + (−0.589 + 2.20i)19-s + (1.96 − 3.40i)23-s + 4.12i·25-s + (4.62 + 2.67i)29-s + (4.82 + 4.82i)31-s + (2.34 − 1.35i)35-s + (2.86 + 10.6i)37-s + (−1.04 − 3.88i)41-s + (6.62 − 3.82i)43-s + (6.84 + 6.84i)47-s + ⋯
L(s)  = 1  + (−0.295 + 0.295i)5-s + (−1.05 − 0.283i)7-s + (1.28 − 0.343i)11-s + (0.471 + 0.881i)13-s + (−0.211 − 0.365i)17-s + (−0.135 + 0.505i)19-s + (0.409 − 0.709i)23-s + 0.825i·25-s + (0.859 + 0.495i)29-s + (0.867 + 0.867i)31-s + (0.396 − 0.229i)35-s + (0.470 + 1.75i)37-s + (−0.162 − 0.606i)41-s + (1.00 − 0.583i)43-s + (0.998 + 0.998i)47-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((0.773−0.634i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s+1/2) L(s)=((0.773−0.634i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 - 0.634i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: 0.773−0.634i0.773 - 0.634i
Analytic conductor: 7.473997.47399
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ936(449,⋅)\chi_{936} (449, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :1/2), 0.773−0.634i)(2,\ 936,\ (\ :1/2),\ 0.773 - 0.634i)

Particular Values

L(1)L(1) ≈\approx 1.29579+0.463579i1.29579 + 0.463579i
L(12)L(\frac12) ≈\approx 1.29579+0.463579i1.29579 + 0.463579i
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
3 1 1
13 1+(−1.70−3.17i)T 1 + (-1.70 - 3.17i)T
good5 1+(0.661−0.661i)T−5iT2 1 + (0.661 - 0.661i)T - 5iT^{2}
7 1+(2.79+0.750i)T+(6.06+3.5i)T2 1 + (2.79 + 0.750i)T + (6.06 + 3.5i)T^{2}
11 1+(−4.25+1.13i)T+(9.52−5.5i)T2 1 + (-4.25 + 1.13i)T + (9.52 - 5.5i)T^{2}
17 1+(0.870+1.50i)T+(−8.5+14.7i)T2 1 + (0.870 + 1.50i)T + (-8.5 + 14.7i)T^{2}
19 1+(0.589−2.20i)T+(−16.4−9.5i)T2 1 + (0.589 - 2.20i)T + (-16.4 - 9.5i)T^{2}
23 1+(−1.96+3.40i)T+(−11.5−19.9i)T2 1 + (-1.96 + 3.40i)T + (-11.5 - 19.9i)T^{2}
29 1+(−4.62−2.67i)T+(14.5+25.1i)T2 1 + (-4.62 - 2.67i)T + (14.5 + 25.1i)T^{2}
31 1+(−4.82−4.82i)T+31iT2 1 + (-4.82 - 4.82i)T + 31iT^{2}
37 1+(−2.86−10.6i)T+(−32.0+18.5i)T2 1 + (-2.86 - 10.6i)T + (-32.0 + 18.5i)T^{2}
41 1+(1.04+3.88i)T+(−35.5+20.5i)T2 1 + (1.04 + 3.88i)T + (-35.5 + 20.5i)T^{2}
43 1+(−6.62+3.82i)T+(21.5−37.2i)T2 1 + (-6.62 + 3.82i)T + (21.5 - 37.2i)T^{2}
47 1+(−6.84−6.84i)T+47iT2 1 + (-6.84 - 6.84i)T + 47iT^{2}
53 1−8.89iT−53T2 1 - 8.89iT - 53T^{2}
59 1+(−1.63+6.08i)T+(−51.0−29.5i)T2 1 + (-1.63 + 6.08i)T + (-51.0 - 29.5i)T^{2}
61 1+(2.39+4.15i)T+(−30.5+52.8i)T2 1 + (2.39 + 4.15i)T + (-30.5 + 52.8i)T^{2}
67 1+(9.52−2.55i)T+(58.0−33.5i)T2 1 + (9.52 - 2.55i)T + (58.0 - 33.5i)T^{2}
71 1+(2.88+0.771i)T+(61.4+35.5i)T2 1 + (2.88 + 0.771i)T + (61.4 + 35.5i)T^{2}
73 1+(0.315−0.315i)T−73iT2 1 + (0.315 - 0.315i)T - 73iT^{2}
79 1+1.75T+79T2 1 + 1.75T + 79T^{2}
83 1+(−9.41+9.41i)T−83iT2 1 + (-9.41 + 9.41i)T - 83iT^{2}
89 1+(−12.6+3.40i)T+(77.0−44.5i)T2 1 + (-12.6 + 3.40i)T + (77.0 - 44.5i)T^{2}
97 1+(1.85−6.91i)T+(−84.0−48.5i)T2 1 + (1.85 - 6.91i)T + (-84.0 - 48.5i)T^{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

−10.17563475184700899862078287199, −9.170774016672123199752214687705, −8.758035353419359508527261193952, −7.45687011853990655826260151259, −6.54531133185340228053308355867, −6.26996205615264666348034439590, −4.67514833716136737959276498930, −3.76251647459368676739284365776, −2.93253322406729663656339954880, −1.19263927847696726227444447487, 0.792403382045317190891647891596, 2.52937589748004598775113026257, 3.67625204633903732110027962427, 4.47607695828676785598426958631, 5.86267878411726954592086857836, 6.41362148428183627606316333769, 7.40807777249822326676154708612, 8.422871698388466896229442709095, 9.172119416999286594953829310530, 9.833533250900768816930129656504

Graph of the ZZ-function along the critical line