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Properties

Label 2-936-39.11-c1-0-2
Degree 22
Conductor 936936
Sign −0.741−0.670i-0.741 - 0.670i
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  + (−1.79 + 1.79i)5-s + (1.02 + 3.83i)7-s + (1.05 − 3.95i)11-s + (1.54 + 3.25i)13-s + (1.68 − 2.91i)17-s + (−7.63 + 2.04i)19-s + (1.54 + 2.67i)23-s − 1.41i·25-s + (−7.02 + 4.05i)29-s + (0.618 + 0.618i)31-s + (−8.70 − 5.02i)35-s + (−8.92 − 2.39i)37-s + (−6.25 − 1.67i)41-s + (8.40 + 4.85i)43-s + (−4.37 − 4.37i)47-s + ⋯
L(s)  = 1  + (−0.801 + 0.801i)5-s + (0.388 + 1.44i)7-s + (0.319 − 1.19i)11-s + (0.428 + 0.903i)13-s + (0.408 − 0.707i)17-s + (−1.75 + 0.469i)19-s + (0.322 + 0.558i)23-s − 0.283i·25-s + (−1.30 + 0.753i)29-s + (0.111 + 0.111i)31-s + (−1.47 − 0.849i)35-s + (−1.46 − 0.392i)37-s + (−0.977 − 0.261i)41-s + (1.28 + 0.740i)43-s + (−0.637 − 0.637i)47-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((−0.741−0.670i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 - 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s+1/2) L(s)=((−0.741−0.670i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.741 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: −0.741−0.670i-0.741 - 0.670i
Analytic conductor: 7.473997.47399
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ936(89,⋅)\chi_{936} (89, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :1/2), −0.741−0.670i)(2,\ 936,\ (\ :1/2),\ -0.741 - 0.670i)

Particular Values

L(1)L(1) ≈\approx 0.348128+0.903530i0.348128 + 0.903530i
L(12)L(\frac12) ≈\approx 0.348128+0.903530i0.348128 + 0.903530i
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+(−1.54−3.25i)T 1 + (-1.54 - 3.25i)T
good5 1+(1.79−1.79i)T−5iT2 1 + (1.79 - 1.79i)T - 5iT^{2}
7 1+(−1.02−3.83i)T+(−6.06+3.5i)T2 1 + (-1.02 - 3.83i)T + (-6.06 + 3.5i)T^{2}
11 1+(−1.05+3.95i)T+(−9.52−5.5i)T2 1 + (-1.05 + 3.95i)T + (-9.52 - 5.5i)T^{2}
17 1+(−1.68+2.91i)T+(−8.5−14.7i)T2 1 + (-1.68 + 2.91i)T + (-8.5 - 14.7i)T^{2}
19 1+(7.63−2.04i)T+(16.4−9.5i)T2 1 + (7.63 - 2.04i)T + (16.4 - 9.5i)T^{2}
23 1+(−1.54−2.67i)T+(−11.5+19.9i)T2 1 + (-1.54 - 2.67i)T + (-11.5 + 19.9i)T^{2}
29 1+(7.02−4.05i)T+(14.5−25.1i)T2 1 + (7.02 - 4.05i)T + (14.5 - 25.1i)T^{2}
31 1+(−0.618−0.618i)T+31iT2 1 + (-0.618 - 0.618i)T + 31iT^{2}
37 1+(8.92+2.39i)T+(32.0+18.5i)T2 1 + (8.92 + 2.39i)T + (32.0 + 18.5i)T^{2}
41 1+(6.25+1.67i)T+(35.5+20.5i)T2 1 + (6.25 + 1.67i)T + (35.5 + 20.5i)T^{2}
43 1+(−8.40−4.85i)T+(21.5+37.2i)T2 1 + (-8.40 - 4.85i)T + (21.5 + 37.2i)T^{2}
47 1+(4.37+4.37i)T+47iT2 1 + (4.37 + 4.37i)T + 47iT^{2}
53 1−13.8iT−53T2 1 - 13.8iT - 53T^{2}
59 1+(−4.03+1.07i)T+(51.0−29.5i)T2 1 + (-4.03 + 1.07i)T + (51.0 - 29.5i)T^{2}
61 1+(4.06−7.04i)T+(−30.5−52.8i)T2 1 + (4.06 - 7.04i)T + (-30.5 - 52.8i)T^{2}
67 1+(−3.17+11.8i)T+(−58.0−33.5i)T2 1 + (-3.17 + 11.8i)T + (-58.0 - 33.5i)T^{2}
71 1+(0.166+0.622i)T+(−61.4+35.5i)T2 1 + (0.166 + 0.622i)T + (-61.4 + 35.5i)T^{2}
73 1+(0.788−0.788i)T−73iT2 1 + (0.788 - 0.788i)T - 73iT^{2}
79 1−15.1T+79T2 1 - 15.1T + 79T^{2}
83 1+(0.0917−0.0917i)T−83iT2 1 + (0.0917 - 0.0917i)T - 83iT^{2}
89 1+(3.46−12.9i)T+(−77.0−44.5i)T2 1 + (3.46 - 12.9i)T + (-77.0 - 44.5i)T^{2}
97 1+(−6.54+1.75i)T+(84.0−48.5i)T2 1 + (-6.54 + 1.75i)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

−10.74513974377557577202365414135, −9.211901354898195561891726977011, −8.801829370349335409038165268251, −7.957995140916279522610249835408, −6.94408874627828536093554645886, −6.09980833043255330775814760084, −5.27472600223900550694240464912, −3.92596618385649394068416768098, −3.11672193062360462970470291728, −1.87913193452879943882823421960, 0.45266864731144251107474444228, 1.81881203101676438935566950469, 3.73016761455899966548572656429, 4.26599766876958418708150808893, 5.07199824754716557258195332081, 6.48433571168777844625766789744, 7.31532087928066485096598002110, 8.101468165449177755155623632927, 8.654957957769500940042322858396, 9.912179183693108680135248115401

Graph of the ZZ-function along the critical line