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Properties

Label 2-2646-9.7-c1-0-10
Degree 22
Conductor 26462646
Sign −0.0871−0.996i-0.0871 - 0.996i
Analytic cond. 21.128421.1284
Root an. cond. 4.596564.59656
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 + (−0.499 − 0.866i)4-s + (0.258 + 0.448i)5-s + 0.999·8-s − 0.517·10-s + (−0.732 + 1.26i)11-s + (1.22 + 2.12i)13-s + (−0.5 + 0.866i)16-s + 3.48·17-s + 0.517·19-s + (0.258 − 0.448i)20-s + (−0.732 − 1.26i)22-s + (3.96 + 6.86i)23-s + (2.36 − 4.09i)25-s − 2.44·26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.115 + 0.200i)5-s + 0.353·8-s − 0.163·10-s + (−0.220 + 0.382i)11-s + (0.339 + 0.588i)13-s + (−0.125 + 0.216i)16-s + 0.845·17-s + 0.118·19-s + (0.0578 − 0.100i)20-s + (−0.156 − 0.270i)22-s + (0.826 + 1.43i)23-s + (0.473 − 0.819i)25-s − 0.480·26-s + ⋯

Functional equation

Λ(s)=(2646s/2 ΓC(s) L(s)=((−0.0871−0.996i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0871 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2646s/2 ΓC(s+1/2) L(s)=((−0.0871−0.996i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0871 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26462646    =    2⋅33⋅722 \cdot 3^{3} \cdot 7^{2}
Sign: −0.0871−0.996i-0.0871 - 0.996i
Analytic conductor: 21.128421.1284
Root analytic conductor: 4.596564.59656
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2646(883,⋅)\chi_{2646} (883, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2646, ( :1/2), −0.0871−0.996i)(2,\ 2646,\ (\ :1/2),\ -0.0871 - 0.996i)

Particular Values

L(1)L(1) ≈\approx 1.4297923261.429792326
L(12)L(\frac12) ≈\approx 1.4297923261.429792326
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
3 1 1
7 1 1
good5 1+(−0.258−0.448i)T+(−2.5+4.33i)T2 1 + (-0.258 - 0.448i)T + (-2.5 + 4.33i)T^{2}
11 1+(0.732−1.26i)T+(−5.5−9.52i)T2 1 + (0.732 - 1.26i)T + (-5.5 - 9.52i)T^{2}
13 1+(−1.22−2.12i)T+(−6.5+11.2i)T2 1 + (-1.22 - 2.12i)T + (-6.5 + 11.2i)T^{2}
17 1−3.48T+17T2 1 - 3.48T + 17T^{2}
19 1−0.517T+19T2 1 - 0.517T + 19T^{2}
23 1+(−3.96−6.86i)T+(−11.5+19.9i)T2 1 + (-3.96 - 6.86i)T + (-11.5 + 19.9i)T^{2}
29 1+(−1.36+2.36i)T+(−14.5−25.1i)T2 1 + (-1.36 + 2.36i)T + (-14.5 - 25.1i)T^{2}
31 1+(3.67+6.36i)T+(−15.5+26.8i)T2 1 + (3.67 + 6.36i)T + (-15.5 + 26.8i)T^{2}
37 1+8T+37T2 1 + 8T + 37T^{2}
41 1+(−2.82−4.89i)T+(−20.5+35.5i)T2 1 + (-2.82 - 4.89i)T + (-20.5 + 35.5i)T^{2}
43 1+(−6.09+10.5i)T+(−21.5−37.2i)T2 1 + (-6.09 + 10.5i)T + (-21.5 - 37.2i)T^{2}
47 1+(2.31−4.00i)T+(−23.5−40.7i)T2 1 + (2.31 - 4.00i)T + (-23.5 - 40.7i)T^{2}
53 1+6.73T+53T2 1 + 6.73T + 53T^{2}
59 1+(−7.39−12.8i)T+(−29.5+51.0i)T2 1 + (-7.39 - 12.8i)T + (-29.5 + 51.0i)T^{2}
61 1+(2.19−3.79i)T+(−30.5−52.8i)T2 1 + (2.19 - 3.79i)T + (-30.5 - 52.8i)T^{2}
67 1+(−1.90−3.29i)T+(−33.5+58.0i)T2 1 + (-1.90 - 3.29i)T + (-33.5 + 58.0i)T^{2}
71 1−0.803T+71T2 1 - 0.803T + 71T^{2}
73 1−4.62T+73T2 1 - 4.62T + 73T^{2}
79 1+(7.06−12.2i)T+(−39.5−68.4i)T2 1 + (7.06 - 12.2i)T + (-39.5 - 68.4i)T^{2}
83 1+(4.94−8.57i)T+(−41.5−71.8i)T2 1 + (4.94 - 8.57i)T + (-41.5 - 71.8i)T^{2}
89 1−16.1T+89T2 1 - 16.1T + 89T^{2}
97 1+(−0.517+0.896i)T+(−48.5−84.0i)T2 1 + (-0.517 + 0.896i)T + (-48.5 - 84.0i)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

−9.063051486017162941692768949138, −8.247084356378315908000078934792, −7.43085083265047523026293199362, −6.96828448545587040373086792994, −5.97114680534973446552773085517, −5.39565484055828799043449480889, −4.41639973142503509726168742615, −3.48217886823819974524648944314, −2.27223498255518410176836390058, −1.08115516723396993399930218732, 0.63924779108701704838973630761, 1.69203459382976939253859355175, 3.00846790339865001767687219059, 3.47107837683834907808804272296, 4.79191853174913357865734027324, 5.35577343986985624906599490495, 6.41285684052566817730929944226, 7.27791376296272286945914777591, 8.101282947201177893458974135789, 8.758815387151240867752788965655

Graph of the ZZ-function along the critical line