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Properties

Label 2-637-91.88-c1-0-31
Degree 22
Conductor 637637
Sign −0.794−0.606i-0.794 - 0.606i
Analytic cond. 5.086475.08647
Root an. cond. 2.255322.25532
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.89 − 1.09i)2-s + 1.79·3-s + (1.39 + 2.41i)4-s + (−1.89 + 1.09i)5-s + (−3.39 − 1.96i)6-s − 1.73i·8-s + 0.208·9-s + 4.79·10-s − 1.27i·11-s + (2.5 + 4.33i)12-s + (−3.5 − 0.866i)13-s + (−3.39 + 1.96i)15-s + (0.895 − 1.55i)16-s + (1.5 + 2.59i)17-s + (−0.395 − 0.228i)18-s − 6.56i·19-s + ⋯
L(s)  = 1  + (−1.34 − 0.773i)2-s + 1.03·3-s + (0.697 + 1.20i)4-s + (−0.847 + 0.489i)5-s + (−1.38 − 0.800i)6-s − 0.612i·8-s + 0.0695·9-s + 1.51·10-s − 0.384i·11-s + (0.721 + 1.24i)12-s + (−0.970 − 0.240i)13-s + (−0.876 + 0.506i)15-s + (0.223 − 0.387i)16-s + (0.363 + 0.630i)17-s + (−0.0932 − 0.0538i)18-s − 1.50i·19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.794−0.606i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.794−0.606i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.794 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.794−0.606i-0.794 - 0.606i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(361,⋅)\chi_{637} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 637, ( :1/2), −0.794−0.606i)(2,\ 637,\ (\ :1/2),\ -0.794 - 0.606i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad7 1 1
13 1+(3.5+0.866i)T 1 + (3.5 + 0.866i)T
good2 1+(1.89+1.09i)T+(1+1.73i)T2 1 + (1.89 + 1.09i)T + (1 + 1.73i)T^{2}
3 1−1.79T+3T2 1 - 1.79T + 3T^{2}
5 1+(1.89−1.09i)T+(2.5−4.33i)T2 1 + (1.89 - 1.09i)T + (2.5 - 4.33i)T^{2}
11 1+1.27iT−11T2 1 + 1.27iT - 11T^{2}
17 1+(−1.5−2.59i)T+(−8.5+14.7i)T2 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2}
19 1+6.56iT−19T2 1 + 6.56iT - 19T^{2}
23 1+(3.79−6.56i)T+(−11.5−19.9i)T2 1 + (3.79 - 6.56i)T + (-11.5 - 19.9i)T^{2}
29 1+(−1.10−1.91i)T+(−14.5+25.1i)T2 1 + (-1.10 - 1.91i)T + (-14.5 + 25.1i)T^{2}
31 1+(7.5+4.33i)T+(15.5+26.8i)T2 1 + (7.5 + 4.33i)T + (15.5 + 26.8i)T^{2}
37 1+(6+3.46i)T+(18.5+32.0i)T2 1 + (6 + 3.46i)T + (18.5 + 32.0i)T^{2}
41 1+(2.20−1.27i)T+(20.5−35.5i)T2 1 + (2.20 - 1.27i)T + (20.5 - 35.5i)T^{2}
43 1+(−2.18+3.78i)T+(−21.5−37.2i)T2 1 + (-2.18 + 3.78i)T + (-21.5 - 37.2i)T^{2}
47 1+(3.70−2.14i)T+(23.5−40.7i)T2 1 + (3.70 - 2.14i)T + (23.5 - 40.7i)T^{2}
53 1+(−6.08+10.5i)T+(−26.5−45.8i)T2 1 + (-6.08 + 10.5i)T + (-26.5 - 45.8i)T^{2}
59 1+(7.66−4.42i)T+(29.5−51.0i)T2 1 + (7.66 - 4.42i)T + (29.5 - 51.0i)T^{2}
61 1+12.7T+61T2 1 + 12.7T + 61T^{2}
67 1−11.4iT−67T2 1 - 11.4iT - 67T^{2}
71 1+(0.791+0.456i)T+(35.5+61.4i)T2 1 + (0.791 + 0.456i)T + (35.5 + 61.4i)T^{2}
73 1+(−3−1.73i)T+(36.5+63.2i)T2 1 + (-3 - 1.73i)T + (36.5 + 63.2i)T^{2}
79 1+(−3−5.19i)T+(−39.5+68.4i)T2 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2}
83 1−3.55iT−83T2 1 - 3.55iT - 83T^{2}
89 1+(2.52+1.45i)T+(44.5+77.0i)T2 1 + (2.52 + 1.45i)T + (44.5 + 77.0i)T^{2}
97 1+(−13.1−7.61i)T+(48.5+84.0i)T2 1 + (-13.1 - 7.61i)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

−9.874856597244581503091418665687, −9.188918583482049933505662723934, −8.487364802641742903168912823373, −7.64677250007074815033225172550, −7.25024355995263600117105028241, −5.45459392373462712278001370540, −3.72838439381942006519464339291, −3.00238638961305347992632021906, −1.96800491764636992644598015215, 0, 1.90064063071080855486757651577, 3.43820993912466889096361815679, 4.62374025012526072344057157228, 6.04682563206607060888512777965, 7.25343191510540273514513274793, 7.82587392553078210127500271384, 8.385808931351345474648786529357, 9.123845962071346004289905342567, 9.849549336082513644982882104581

Graph of the ZZ-function along the critical line