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Properties

Label 2-1176-7.2-c1-0-16
Degree 22
Conductor 11761176
Sign 0.386+0.922i0.386 + 0.922i
Analytic cond. 9.390409.39040
Root an. cond. 3.064373.06437
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  + (0.5 + 0.866i)3-s + (1 − 1.73i)5-s + (−0.499 + 0.866i)9-s + (−2 − 3.46i)11-s − 2·13-s + 1.99·15-s + (−1 − 1.73i)17-s + (2 − 3.46i)19-s + (4 − 6.92i)23-s + (0.500 + 0.866i)25-s − 0.999·27-s + 6·29-s + (−4 − 6.92i)31-s + (1.99 − 3.46i)33-s + (−3 + 5.19i)37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.447 − 0.774i)5-s + (−0.166 + 0.288i)9-s + (−0.603 − 1.04i)11-s − 0.554·13-s + 0.516·15-s + (−0.242 − 0.420i)17-s + (0.458 − 0.794i)19-s + (0.834 − 1.44i)23-s + (0.100 + 0.173i)25-s − 0.192·27-s + 1.11·29-s + (−0.718 − 1.24i)31-s + (0.348 − 0.603i)33-s + (−0.493 + 0.854i)37-s + ⋯

Functional equation

Λ(s)=(1176s/2 ΓC(s) L(s)=((0.386+0.922i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1176s/2 ΓC(s+1/2) L(s)=((0.386+0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 11761176    =    23⋅3⋅722^{3} \cdot 3 \cdot 7^{2}
Sign: 0.386+0.922i0.386 + 0.922i
Analytic conductor: 9.390409.39040
Root analytic conductor: 3.064373.06437
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1176(961,⋅)\chi_{1176} (961, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1176, ( :1/2), 0.386+0.922i)(2,\ 1176,\ (\ :1/2),\ 0.386 + 0.922i)

Particular Values

L(1)L(1) ≈\approx 1.6301726001.630172600
L(12)L(\frac12) ≈\approx 1.6301726001.630172600
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+(−0.5−0.866i)T 1 + (-0.5 - 0.866i)T
7 1 1
good5 1+(−1+1.73i)T+(−2.5−4.33i)T2 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2}
11 1+(2+3.46i)T+(−5.5+9.52i)T2 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2}
13 1+2T+13T2 1 + 2T + 13T^{2}
17 1+(1+1.73i)T+(−8.5+14.7i)T2 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2}
19 1+(−2+3.46i)T+(−9.5−16.4i)T2 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2}
23 1+(−4+6.92i)T+(−11.5−19.9i)T2 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2}
29 1−6T+29T2 1 - 6T + 29T^{2}
31 1+(4+6.92i)T+(−15.5+26.8i)T2 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2}
37 1+(3−5.19i)T+(−18.5−32.0i)T2 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2}
41 1+6T+41T2 1 + 6T + 41T^{2}
43 1−4T+43T2 1 - 4T + 43T^{2}
47 1+(−23.5−40.7i)T2 1 + (-23.5 - 40.7i)T^{2}
53 1+(−1−1.73i)T+(−26.5+45.8i)T2 1 + (-1 - 1.73i)T + (-26.5 + 45.8i)T^{2}
59 1+(2+3.46i)T+(−29.5+51.0i)T2 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2}
61 1+(−1+1.73i)T+(−30.5−52.8i)T2 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2}
67 1+(−2−3.46i)T+(−33.5+58.0i)T2 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2}
71 1−8T+71T2 1 - 8T + 71T^{2}
73 1+(5+8.66i)T+(−36.5+63.2i)T2 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2}
79 1+(−4+6.92i)T+(−39.5−68.4i)T2 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2}
83 1+4T+83T2 1 + 4T + 83T^{2}
89 1+(−3+5.19i)T+(−44.5−77.0i)T2 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2}
97 1−2T+97T2 1 - 2T + 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

−9.483509639300566864530743596231, −8.859029203162157645556683985493, −8.253465339169081962557179281522, −7.19606435561194585456817772159, −6.14364745377578558559126635252, −5.08427308143170178175268646645, −4.71275716493911233208348817303, −3.27481007839388137981136165915, −2.39971869454034995860714342133, −0.67598066245933542679691704562, 1.62060026121176131173092243339, 2.58325468363444107765005096270, 3.53850914002634254838330623309, 4.90602183779018520341297399732, 5.75164118249395832507971187875, 6.90671324548128995301522749854, 7.23974721291906822564795327381, 8.180636641101783764424918200312, 9.166535085941774680575927494016, 10.02426693125771851321324607233

Graph of the ZZ-function along the critical line