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Properties

Label 2-627-19.7-c1-0-20
Degree 22
Conductor 627627
Sign −0.853−0.521i-0.853 - 0.521i
Analytic cond. 5.006625.00662
Root an. cond. 2.237542.23754
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  + (−1.35 − 2.35i)2-s + (−0.5 − 0.866i)3-s + (−2.69 + 4.65i)4-s + (−0.467 − 0.809i)5-s + (−1.35 + 2.35i)6-s + 4.17·7-s + 9.18·8-s + (−0.499 + 0.866i)9-s + (−1.27 + 2.20i)10-s − 11-s + 5.38·12-s + (−0.486 + 0.841i)13-s + (−5.67 − 9.83i)14-s + (−0.467 + 0.809i)15-s + (−7.09 − 12.2i)16-s + (−2.03 − 3.51i)17-s + ⋯
L(s)  = 1  + (−0.960 − 1.66i)2-s + (−0.288 − 0.499i)3-s + (−1.34 + 2.32i)4-s + (−0.209 − 0.362i)5-s + (−0.554 + 0.960i)6-s + 1.57·7-s + 3.24·8-s + (−0.166 + 0.288i)9-s + (−0.401 + 0.695i)10-s − 0.301·11-s + 1.55·12-s + (−0.134 + 0.233i)13-s + (−1.51 − 2.62i)14-s + (−0.120 + 0.209i)15-s + (−1.77 − 3.07i)16-s + (−0.492 − 0.853i)17-s + ⋯

Functional equation

Λ(s)=(627s/2 ΓC(s) L(s)=((−0.853−0.521i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(627s/2 ΓC(s+1/2) L(s)=((−0.853−0.521i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 - 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 627627    =    3⋅11⋅193 \cdot 11 \cdot 19
Sign: −0.853−0.521i-0.853 - 0.521i
Analytic conductor: 5.006625.00662
Root analytic conductor: 2.237542.23754
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ627(463,⋅)\chi_{627} (463, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 627, ( :1/2), −0.853−0.521i)(2,\ 627,\ (\ :1/2),\ -0.853 - 0.521i)

Particular Values

L(1)L(1) ≈\approx 0.173483+0.616406i0.173483 + 0.616406i
L(12)L(\frac12) ≈\approx 0.173483+0.616406i0.173483 + 0.616406i
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)
bad3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
11 1+T 1 + T
19 1+(4.11+1.43i)T 1 + (4.11 + 1.43i)T
good2 1+(1.35+2.35i)T+(−1+1.73i)T2 1 + (1.35 + 2.35i)T + (-1 + 1.73i)T^{2}
5 1+(0.467+0.809i)T+(−2.5+4.33i)T2 1 + (0.467 + 0.809i)T + (-2.5 + 4.33i)T^{2}
7 1−4.17T+7T2 1 - 4.17T + 7T^{2}
13 1+(0.486−0.841i)T+(−6.5−11.2i)T2 1 + (0.486 - 0.841i)T + (-6.5 - 11.2i)T^{2}
17 1+(2.03+3.51i)T+(−8.5+14.7i)T2 1 + (2.03 + 3.51i)T + (-8.5 + 14.7i)T^{2}
23 1+(−4.65+8.06i)T+(−11.5−19.9i)T2 1 + (-4.65 + 8.06i)T + (-11.5 - 19.9i)T^{2}
29 1+(0.0766−0.132i)T+(−14.5−25.1i)T2 1 + (0.0766 - 0.132i)T + (-14.5 - 25.1i)T^{2}
31 1+3.05T+31T2 1 + 3.05T + 31T^{2}
37 1+4.81T+37T2 1 + 4.81T + 37T^{2}
41 1+(2.91+5.05i)T+(−20.5+35.5i)T2 1 + (2.91 + 5.05i)T + (-20.5 + 35.5i)T^{2}
43 1+(−5.79−10.0i)T+(−21.5+37.2i)T2 1 + (-5.79 - 10.0i)T + (-21.5 + 37.2i)T^{2}
47 1+(0.867−1.50i)T+(−23.5−40.7i)T2 1 + (0.867 - 1.50i)T + (-23.5 - 40.7i)T^{2}
53 1+(0.367−0.637i)T+(−26.5−45.8i)T2 1 + (0.367 - 0.637i)T + (-26.5 - 45.8i)T^{2}
59 1+(2.98+5.16i)T+(−29.5+51.0i)T2 1 + (2.98 + 5.16i)T + (-29.5 + 51.0i)T^{2}
61 1+(−2.37+4.11i)T+(−30.5−52.8i)T2 1 + (-2.37 + 4.11i)T + (-30.5 - 52.8i)T^{2}
67 1+(−3.59+6.22i)T+(−33.5−58.0i)T2 1 + (-3.59 + 6.22i)T + (-33.5 - 58.0i)T^{2}
71 1+(−3.90−6.76i)T+(−35.5+61.4i)T2 1 + (-3.90 - 6.76i)T + (-35.5 + 61.4i)T^{2}
73 1+(5.61+9.72i)T+(−36.5+63.2i)T2 1 + (5.61 + 9.72i)T + (-36.5 + 63.2i)T^{2}
79 1+(7.31+12.6i)T+(−39.5+68.4i)T2 1 + (7.31 + 12.6i)T + (-39.5 + 68.4i)T^{2}
83 1+4.35T+83T2 1 + 4.35T + 83T^{2}
89 1+(7.58−13.1i)T+(−44.5−77.0i)T2 1 + (7.58 - 13.1i)T + (-44.5 - 77.0i)T^{2}
97 1+(0.525+0.910i)T+(−48.5+84.0i)T2 1 + (0.525 + 0.910i)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

−10.49226753042344901985530161637, −9.139428848236019520256532048795, −8.542302857346087208568082662199, −7.926086490271844637665717217509, −6.92392865008310109874856724721, −4.87328076062000119498681078889, −4.43896090829304772892719411191, −2.69839233445214445121569845715, −1.81533541244290032747277457749, −0.53892193752777882306174372226, 1.54721344136695604898172448670, 4.10161877799114139805574222748, 5.16312878727932510584672043101, 5.65714728751514833247041291262, 6.92762193503113121110238844617, 7.58298377916991779931009272764, 8.453985600255979263235490814840, 8.983100943489349499560726565239, 10.15467158617182563131755966657, 10.81115314071344968414474544047

Graph of the ZZ-function along the critical line