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Properties

Label 2-1078-7.4-c1-0-11
Degree 22
Conductor 10781078
Sign 0.947−0.318i0.947 - 0.318i
Analytic cond. 8.607878.60787
Root an. cond. 2.933912.93391
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)2-s + (−1.41 + 2.44i)3-s + (−0.499 + 0.866i)4-s + 2.82·6-s + 0.999·8-s + (−2.49 − 4.33i)9-s + (0.5 − 0.866i)11-s + (−1.41 − 2.44i)12-s + 4.24·13-s + (−0.5 − 0.866i)16-s + (1.41 − 2.44i)17-s + (−2.5 + 4.33i)18-s + (2.12 + 3.67i)19-s − 0.999·22-s + (−3 − 5.19i)23-s + (−1.41 + 2.44i)24-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.816 + 1.41i)3-s + (−0.249 + 0.433i)4-s + 1.15·6-s + 0.353·8-s + (−0.833 − 1.44i)9-s + (0.150 − 0.261i)11-s + (−0.408 − 0.707i)12-s + 1.17·13-s + (−0.125 − 0.216i)16-s + (0.342 − 0.594i)17-s + (−0.589 + 1.02i)18-s + (0.486 + 0.842i)19-s − 0.213·22-s + (−0.625 − 1.08i)23-s + (−0.288 + 0.499i)24-s + ⋯

Functional equation

Λ(s)=(1078s/2 ΓC(s) L(s)=((0.947−0.318i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1078s/2 ΓC(s+1/2) L(s)=((0.947−0.318i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10781078    =    2⋅72⋅112 \cdot 7^{2} \cdot 11
Sign: 0.947−0.318i0.947 - 0.318i
Analytic conductor: 8.607878.60787
Root analytic conductor: 2.933912.93391
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1078(67,⋅)\chi_{1078} (67, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1078, ( :1/2), 0.947−0.318i)(2,\ 1078,\ (\ :1/2),\ 0.947 - 0.318i)

Particular Values

L(1)L(1) ≈\approx 0.99199687010.9919968701
L(12)L(\frac12) ≈\approx 0.99199687010.9919968701
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+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
7 1 1
11 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
good3 1+(1.41−2.44i)T+(−1.5−2.59i)T2 1 + (1.41 - 2.44i)T + (-1.5 - 2.59i)T^{2}
5 1+(−2.5+4.33i)T2 1 + (-2.5 + 4.33i)T^{2}
13 1−4.24T+13T2 1 - 4.24T + 13T^{2}
17 1+(−1.41+2.44i)T+(−8.5−14.7i)T2 1 + (-1.41 + 2.44i)T + (-8.5 - 14.7i)T^{2}
19 1+(−2.12−3.67i)T+(−9.5+16.4i)T2 1 + (-2.12 - 3.67i)T + (-9.5 + 16.4i)T^{2}
23 1+(3+5.19i)T+(−11.5+19.9i)T2 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2}
29 1+4T+29T2 1 + 4T + 29T^{2}
31 1+(−3.53+6.12i)T+(−15.5−26.8i)T2 1 + (-3.53 + 6.12i)T + (-15.5 - 26.8i)T^{2}
37 1+(1+1.73i)T+(−18.5+32.0i)T2 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2}
41 1−2.82T+41T2 1 - 2.82T + 41T^{2}
43 1−10T+43T2 1 - 10T + 43T^{2}
47 1+(−6.36−11.0i)T+(−23.5+40.7i)T2 1 + (-6.36 - 11.0i)T + (-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+(5.65−9.79i)T+(−29.5−51.0i)T2 1 + (5.65 - 9.79i)T + (-29.5 - 51.0i)T^{2}
61 1+(−4.94−8.57i)T+(−30.5+52.8i)T2 1 + (-4.94 - 8.57i)T + (-30.5 + 52.8i)T^{2}
67 1+(4−6.92i)T+(−33.5−58.0i)T2 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2}
71 1−16T+71T2 1 - 16T + 71T^{2}
73 1+(−4.24+7.34i)T+(−36.5−63.2i)T2 1 + (-4.24 + 7.34i)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+12.7T+83T2 1 + 12.7T + 83T^{2}
89 1+(3.53+6.12i)T+(−44.5+77.0i)T2 1 + (3.53 + 6.12i)T + (-44.5 + 77.0i)T^{2}
97 1+7.07T+97T2 1 + 7.07T + 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

−10.04782721068340521854276309270, −9.356936636813252356059572395977, −8.625616295952235757414077206119, −7.64052900021886298656851426515, −6.16565291277193775863964397411, −5.65331695370207060271284626456, −4.36382061921744176456898733077, −3.93916196950605438771150927825, −2.74174831006288833916842200838, −0.835246760862546430766975005413, 0.905456057023445651646997909672, 1.85187184812690158034404391217, 3.60119681789291084263510779006, 5.13649040754469950502269705972, 5.81593142170049109768435747105, 6.58264945385010845091290977510, 7.23578761412582433242552160222, 7.940360557518477234317406468495, 8.784229410582820689897826088916, 9.714211581038556159289075838540

Graph of the ZZ-function along the critical line