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Properties

Label 2-936-936.517-c1-0-108
Degree 22
Conductor 936936
Sign 0.434+0.900i0.434 + 0.900i
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  + (−1.32 − 0.487i)2-s + (1.69 − 0.373i)3-s + (1.52 + 1.29i)4-s + (1.73 − 3.00i)5-s + (−2.42 − 0.328i)6-s + 2.47i·7-s + (−1.39 − 2.46i)8-s + (2.72 − 1.26i)9-s + (−3.76 + 3.14i)10-s + (−1.61 + 2.79i)11-s + (3.06 + 1.61i)12-s + (−3.34 + 1.35i)13-s + (1.20 − 3.29i)14-s + (1.81 − 5.72i)15-s + (0.653 + 3.94i)16-s + (2.89 − 5.01i)17-s + ⋯
L(s)  = 1  + (−0.938 − 0.344i)2-s + (0.976 − 0.215i)3-s + (0.762 + 0.646i)4-s + (0.775 − 1.34i)5-s + (−0.990 − 0.134i)6-s + 0.936i·7-s + (−0.493 − 0.869i)8-s + (0.907 − 0.420i)9-s + (−1.19 + 0.993i)10-s + (−0.487 + 0.843i)11-s + (0.884 + 0.467i)12-s + (−0.926 + 0.375i)13-s + (0.322 − 0.879i)14-s + (0.467 − 1.47i)15-s + (0.163 + 0.986i)16-s + (0.701 − 1.21i)17-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((0.434+0.900i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s+1/2) L(s)=((0.434+0.900i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: 0.434+0.900i0.434 + 0.900i
Analytic conductor: 7.473997.47399
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ936(517,⋅)\chi_{936} (517, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :1/2), 0.434+0.900i)(2,\ 936,\ (\ :1/2),\ 0.434 + 0.900i)

Particular Values

L(1)L(1) ≈\approx 1.43133−0.898666i1.43133 - 0.898666i
L(12)L(\frac12) ≈\approx 1.43133−0.898666i1.43133 - 0.898666i
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+(1.32+0.487i)T 1 + (1.32 + 0.487i)T
3 1+(−1.69+0.373i)T 1 + (-1.69 + 0.373i)T
13 1+(3.34−1.35i)T 1 + (3.34 - 1.35i)T
good5 1+(−1.73+3.00i)T+(−2.5−4.33i)T2 1 + (-1.73 + 3.00i)T + (-2.5 - 4.33i)T^{2}
7 1−2.47iT−7T2 1 - 2.47iT - 7T^{2}
11 1+(1.61−2.79i)T+(−5.5−9.52i)T2 1 + (1.61 - 2.79i)T + (-5.5 - 9.52i)T^{2}
17 1+(−2.89+5.01i)T+(−8.5−14.7i)T2 1 + (-2.89 + 5.01i)T + (-8.5 - 14.7i)T^{2}
19 1+(−2.26+3.92i)T+(−9.5−16.4i)T2 1 + (-2.26 + 3.92i)T + (-9.5 - 16.4i)T^{2}
23 1−9.31T+23T2 1 - 9.31T + 23T^{2}
29 1+(3.40+1.96i)T+(14.5+25.1i)T2 1 + (3.40 + 1.96i)T + (14.5 + 25.1i)T^{2}
31 1+(−5.59−3.23i)T+(15.5+26.8i)T2 1 + (-5.59 - 3.23i)T + (15.5 + 26.8i)T^{2}
37 1+(1.66+2.87i)T+(−18.5+32.0i)T2 1 + (1.66 + 2.87i)T + (-18.5 + 32.0i)T^{2}
41 1+2.56iT−41T2 1 + 2.56iT - 41T^{2}
43 1+9.40iT−43T2 1 + 9.40iT - 43T^{2}
47 1+(7.93−4.57i)T+(23.5−40.7i)T2 1 + (7.93 - 4.57i)T + (23.5 - 40.7i)T^{2}
53 1+0.880iT−53T2 1 + 0.880iT - 53T^{2}
59 1+(−4.67−8.09i)T+(−29.5+51.0i)T2 1 + (-4.67 - 8.09i)T + (-29.5 + 51.0i)T^{2}
61 1−12.6iT−61T2 1 - 12.6iT - 61T^{2}
67 1+3.65T+67T2 1 + 3.65T + 67T^{2}
71 1+(3.31+1.91i)T+(35.5+61.4i)T2 1 + (3.31 + 1.91i)T + (35.5 + 61.4i)T^{2}
73 1−6.92iT−73T2 1 - 6.92iT - 73T^{2}
79 1+(5.76+9.99i)T+(−39.5+68.4i)T2 1 + (5.76 + 9.99i)T + (-39.5 + 68.4i)T^{2}
83 1+(1.07+1.86i)T+(−41.5+71.8i)T2 1 + (1.07 + 1.86i)T + (-41.5 + 71.8i)T^{2}
89 1+(1.67−0.964i)T+(44.5−77.0i)T2 1 + (1.67 - 0.964i)T + (44.5 - 77.0i)T^{2}
97 1−5.53iT−97T2 1 - 5.53iT - 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

−9.550286548385151835811706696527, −9.088987661161376868761418268954, −8.681586515020954770600510034464, −7.48077040965802014854332265496, −7.01040290588439635361953595146, −5.40939807609641499522197616645, −4.67209280832052748577046586245, −2.88232555875016317050562788605, −2.26920103125289776549597752972, −1.08629804717760564042892262567, 1.46478118585463904424441606733, 2.79599798944543426730691111469, 3.37957626673582434836735811855, 5.15264707265935279447055567600, 6.24885429073123646576956806240, 7.06493655858251159553227894951, 7.75691997805107698673856682838, 8.378465636182048336616705988597, 9.704329892686110152046446505937, 9.964704638453795236912296838263

Graph of the ZZ-function along the critical line