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Properties

Label 2-2664-37.26-c1-0-25
Degree 22
Conductor 26642664
Sign 0.993−0.116i0.993 - 0.116i
Analytic cond. 21.272121.2721
Root an. cond. 4.612174.61217
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  + (0.717 − 1.24i)5-s + (−0.0542 + 0.0938i)7-s + 4.03·11-s + (−1.19 + 2.07i)13-s + (−0.5 − 0.866i)17-s + (−1.82 + 3.15i)19-s + 3.53·23-s + (1.46 + 2.54i)25-s − 1.30·29-s + 4.03·31-s + (0.0778 + 0.134i)35-s + (5.87 − 1.58i)37-s + (−1.53 + 2.65i)41-s + 2.43·43-s − 5.40·47-s + ⋯
L(s)  = 1  + (0.321 − 0.556i)5-s + (−0.0204 + 0.0354i)7-s + 1.21·11-s + (−0.332 + 0.575i)13-s + (−0.121 − 0.210i)17-s + (−0.417 + 0.723i)19-s + 0.736·23-s + (0.293 + 0.509i)25-s − 0.242·29-s + 0.725·31-s + (0.0131 + 0.0227i)35-s + (0.965 − 0.260i)37-s + (−0.239 + 0.414i)41-s + 0.371·43-s − 0.788·47-s + ⋯

Functional equation

Λ(s)=(2664s/2 ΓC(s) L(s)=((0.993−0.116i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2664s/2 ΓC(s+1/2) L(s)=((0.993−0.116i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26642664    =    23⋅32⋅372^{3} \cdot 3^{2} \cdot 37
Sign: 0.993−0.116i0.993 - 0.116i
Analytic conductor: 21.272121.2721
Root analytic conductor: 4.612174.61217
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2664(433,⋅)\chi_{2664} (433, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2664, ( :1/2), 0.993−0.116i)(2,\ 2664,\ (\ :1/2),\ 0.993 - 0.116i)

Particular Values

L(1)L(1) ≈\approx 2.0667352342.066735234
L(12)L(\frac12) ≈\approx 2.0667352342.066735234
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
37 1+(−5.87+1.58i)T 1 + (-5.87 + 1.58i)T
good5 1+(−0.717+1.24i)T+(−2.5−4.33i)T2 1 + (-0.717 + 1.24i)T + (-2.5 - 4.33i)T^{2}
7 1+(0.0542−0.0938i)T+(−3.5−6.06i)T2 1 + (0.0542 - 0.0938i)T + (-3.5 - 6.06i)T^{2}
11 1−4.03T+11T2 1 - 4.03T + 11T^{2}
13 1+(1.19−2.07i)T+(−6.5−11.2i)T2 1 + (1.19 - 2.07i)T + (-6.5 - 11.2i)T^{2}
17 1+(0.5+0.866i)T+(−8.5+14.7i)T2 1 + (0.5 + 0.866i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.82−3.15i)T+(−9.5−16.4i)T2 1 + (1.82 - 3.15i)T + (-9.5 - 16.4i)T^{2}
23 1−3.53T+23T2 1 - 3.53T + 23T^{2}
29 1+1.30T+29T2 1 + 1.30T + 29T^{2}
31 1−4.03T+31T2 1 - 4.03T + 31T^{2}
41 1+(1.53−2.65i)T+(−20.5−35.5i)T2 1 + (1.53 - 2.65i)T + (-20.5 - 35.5i)T^{2}
43 1−2.43T+43T2 1 - 2.43T + 43T^{2}
47 1+5.40T+47T2 1 + 5.40T + 47T^{2}
53 1+(−3.91−6.77i)T+(−26.5+45.8i)T2 1 + (-3.91 - 6.77i)T + (-26.5 + 45.8i)T^{2}
59 1+(3.83+6.65i)T+(−29.5+51.0i)T2 1 + (3.83 + 6.65i)T + (-29.5 + 51.0i)T^{2}
61 1+(−1.56+2.70i)T+(−30.5−52.8i)T2 1 + (-1.56 + 2.70i)T + (-30.5 - 52.8i)T^{2}
67 1+(−0.614+1.06i)T+(−33.5−58.0i)T2 1 + (-0.614 + 1.06i)T + (-33.5 - 58.0i)T^{2}
71 1+(1.69−2.93i)T+(−35.5−61.4i)T2 1 + (1.69 - 2.93i)T + (-35.5 - 61.4i)T^{2}
73 1−5.76T+73T2 1 - 5.76T + 73T^{2}
79 1+(−6.34+10.9i)T+(−39.5−68.4i)T2 1 + (-6.34 + 10.9i)T + (-39.5 - 68.4i)T^{2}
83 1+(1.01+1.76i)T+(−41.5+71.8i)T2 1 + (1.01 + 1.76i)T + (-41.5 + 71.8i)T^{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+13.0T+97T2 1 + 13.0T + 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

−9.119414757375985689628766872528, −8.195544272928135526710673500826, −7.30406160652861548828511115196, −6.51348094658998381087607636189, −5.87062056030465496083513753812, −4.83523892866618591543243806711, −4.22685974463000271996488250424, −3.19880897799181487032782769271, −1.97550276624280519252539574730, −1.03623725306129079246591329151, 0.853297674992731000922790301080, 2.19778040504948655645376661942, 3.05590644663491427302259916187, 4.04344091963660453837423640205, 4.88735476966402852858122524619, 5.88165903598750486991685849000, 6.65343891708031478354128232466, 7.08389630374652636247454285507, 8.167928728179624251217929255172, 8.846350460247597462101170076423

Graph of the ZZ-function along the critical line