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Properties

Label 2-936-39.2-c1-0-1
Degree 22
Conductor 936936
Sign −0.473−0.880i-0.473 - 0.880i
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  + (−2.65 − 2.65i)5-s + (1.47 − 0.396i)7-s + (−4.40 − 1.18i)11-s + (−2.78 + 2.29i)13-s + (−2.30 + 3.98i)17-s + (0.611 + 2.28i)19-s + (3.26 + 5.64i)23-s + 9.11i·25-s + (6.53 − 3.77i)29-s + (3.52 − 3.52i)31-s + (−4.98 − 2.87i)35-s + (−1.56 + 5.84i)37-s + (0.0148 − 0.0553i)41-s + (−2.57 − 1.48i)43-s + (−7.10 + 7.10i)47-s + ⋯
L(s)  = 1  + (−1.18 − 1.18i)5-s + (0.559 − 0.149i)7-s + (−1.32 − 0.356i)11-s + (−0.771 + 0.636i)13-s + (−0.557 + 0.966i)17-s + (0.140 + 0.523i)19-s + (0.679 + 1.17i)23-s + 1.82i·25-s + (1.21 − 0.701i)29-s + (0.633 − 0.633i)31-s + (−0.842 − 0.486i)35-s + (−0.257 + 0.961i)37-s + (0.00231 − 0.00863i)41-s + (−0.393 − 0.226i)43-s + (−1.03 + 1.03i)47-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((−0.473−0.880i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s+1/2) L(s)=((−0.473−0.880i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.473 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: −0.473−0.880i-0.473 - 0.880i
Analytic conductor: 7.473997.47399
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ936(665,⋅)\chi_{936} (665, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :1/2), −0.473−0.880i)(2,\ 936,\ (\ :1/2),\ -0.473 - 0.880i)

Particular Values

L(1)L(1) ≈\approx 0.157350+0.263387i0.157350 + 0.263387i
L(12)L(\frac12) ≈\approx 0.157350+0.263387i0.157350 + 0.263387i
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+(2.78−2.29i)T 1 + (2.78 - 2.29i)T
good5 1+(2.65+2.65i)T+5iT2 1 + (2.65 + 2.65i)T + 5iT^{2}
7 1+(−1.47+0.396i)T+(6.06−3.5i)T2 1 + (-1.47 + 0.396i)T + (6.06 - 3.5i)T^{2}
11 1+(4.40+1.18i)T+(9.52+5.5i)T2 1 + (4.40 + 1.18i)T + (9.52 + 5.5i)T^{2}
17 1+(2.30−3.98i)T+(−8.5−14.7i)T2 1 + (2.30 - 3.98i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.611−2.28i)T+(−16.4+9.5i)T2 1 + (-0.611 - 2.28i)T + (-16.4 + 9.5i)T^{2}
23 1+(−3.26−5.64i)T+(−11.5+19.9i)T2 1 + (-3.26 - 5.64i)T + (-11.5 + 19.9i)T^{2}
29 1+(−6.53+3.77i)T+(14.5−25.1i)T2 1 + (-6.53 + 3.77i)T + (14.5 - 25.1i)T^{2}
31 1+(−3.52+3.52i)T−31iT2 1 + (-3.52 + 3.52i)T - 31iT^{2}
37 1+(1.56−5.84i)T+(−32.0−18.5i)T2 1 + (1.56 - 5.84i)T + (-32.0 - 18.5i)T^{2}
41 1+(−0.0148+0.0553i)T+(−35.5−20.5i)T2 1 + (-0.0148 + 0.0553i)T + (-35.5 - 20.5i)T^{2}
43 1+(2.57+1.48i)T+(21.5+37.2i)T2 1 + (2.57 + 1.48i)T + (21.5 + 37.2i)T^{2}
47 1+(7.10−7.10i)T−47iT2 1 + (7.10 - 7.10i)T - 47iT^{2}
53 1−8.48iT−53T2 1 - 8.48iT - 53T^{2}
59 1+(2.84+10.6i)T+(−51.0+29.5i)T2 1 + (2.84 + 10.6i)T + (-51.0 + 29.5i)T^{2}
61 1+(0.615−1.06i)T+(−30.5−52.8i)T2 1 + (0.615 - 1.06i)T + (-30.5 - 52.8i)T^{2}
67 1+(8.23+2.20i)T+(58.0+33.5i)T2 1 + (8.23 + 2.20i)T + (58.0 + 33.5i)T^{2}
71 1+(12.9−3.46i)T+(61.4−35.5i)T2 1 + (12.9 - 3.46i)T + (61.4 - 35.5i)T^{2}
73 1+(−8.01−8.01i)T+73iT2 1 + (-8.01 - 8.01i)T + 73iT^{2}
79 1+12.4T+79T2 1 + 12.4T + 79T^{2}
83 1+(9.98+9.98i)T+83iT2 1 + (9.98 + 9.98i)T + 83iT^{2}
89 1+(−0.222−0.0596i)T+(77.0+44.5i)T2 1 + (-0.222 - 0.0596i)T + (77.0 + 44.5i)T^{2}
97 1+(2.90+10.8i)T+(−84.0+48.5i)T2 1 + (2.90 + 10.8i)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

−10.35768323198949693947443475047, −9.417958401605798280823698763450, −8.288856599673766186847239805818, −8.113837014778610545800705016177, −7.23032435774558036168162806614, −5.86510799062719165412684425288, −4.75924735501910634674974808225, −4.42084698872520432842504392794, −3.06658090851243117207296608657, −1.45903954056284834626876222292, 0.14504860070980349765094815645, 2.58063483372013207819488003851, 3.05990166339890862057166331651, 4.60500166754350479594516390358, 5.11613287577145948707939285593, 6.72414805246658522523536246189, 7.23285401997291221884523277376, 7.990637844896354260491271137119, 8.720690316582211631338316124591, 10.15158284881804748568486559937

Graph of the ZZ-function along the critical line