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Properties

Label 2-627-19.11-c1-0-0
Degree 22
Conductor 627627
Sign 0.327+0.944i0.327 + 0.944i
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  + (−0.794 + 1.37i)2-s + (−0.5 + 0.866i)3-s + (−0.263 − 0.455i)4-s + (−1.98 + 3.44i)5-s + (−0.794 − 1.37i)6-s + 0.0905·7-s − 2.34·8-s + (−0.499 − 0.866i)9-s + (−3.16 − 5.47i)10-s − 11-s + 0.526·12-s + (−1.02 − 1.77i)13-s + (−0.0719 + 0.124i)14-s + (−1.98 − 3.44i)15-s + (2.38 − 4.13i)16-s + (−0.207 + 0.359i)17-s + ⋯
L(s)  = 1  + (−0.561 + 0.973i)2-s + (−0.288 + 0.499i)3-s + (−0.131 − 0.227i)4-s + (−0.889 + 1.54i)5-s + (−0.324 − 0.561i)6-s + 0.0342·7-s − 0.828·8-s + (−0.166 − 0.288i)9-s + (−0.999 − 1.73i)10-s − 0.301·11-s + 0.151·12-s + (−0.284 − 0.492i)13-s + (−0.0192 + 0.0333i)14-s + (−0.513 − 0.889i)15-s + (0.596 − 1.03i)16-s + (−0.0503 + 0.0871i)17-s + ⋯

Functional equation

Λ(s)=(627s/2 ΓC(s) L(s)=((0.327+0.944i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(627s/2 ΓC(s+1/2) L(s)=((0.327+0.944i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 627627    =    3⋅11⋅193 \cdot 11 \cdot 19
Sign: 0.327+0.944i0.327 + 0.944i
Analytic conductor: 5.006625.00662
Root analytic conductor: 2.237542.23754
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ627(562,⋅)\chi_{627} (562, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 627, ( :1/2), 0.327+0.944i)(2,\ 627,\ (\ :1/2),\ 0.327 + 0.944i)

Particular Values

L(1)L(1) ≈\approx 0.265939−0.189340i0.265939 - 0.189340i
L(12)L(\frac12) ≈\approx 0.265939−0.189340i0.265939 - 0.189340i
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+(−0.524−4.32i)T 1 + (-0.524 - 4.32i)T
good2 1+(0.794−1.37i)T+(−1−1.73i)T2 1 + (0.794 - 1.37i)T + (-1 - 1.73i)T^{2}
5 1+(1.98−3.44i)T+(−2.5−4.33i)T2 1 + (1.98 - 3.44i)T + (-2.5 - 4.33i)T^{2}
7 1−0.0905T+7T2 1 - 0.0905T + 7T^{2}
13 1+(1.02+1.77i)T+(−6.5+11.2i)T2 1 + (1.02 + 1.77i)T + (-6.5 + 11.2i)T^{2}
17 1+(0.207−0.359i)T+(−8.5−14.7i)T2 1 + (0.207 - 0.359i)T + (-8.5 - 14.7i)T^{2}
23 1+(−0.0351−0.0608i)T+(−11.5+19.9i)T2 1 + (-0.0351 - 0.0608i)T + (-11.5 + 19.9i)T^{2}
29 1+(−3.91−6.77i)T+(−14.5+25.1i)T2 1 + (-3.91 - 6.77i)T + (-14.5 + 25.1i)T^{2}
31 1−1.40T+31T2 1 - 1.40T + 31T^{2}
37 1−7.87T+37T2 1 - 7.87T + 37T^{2}
41 1+(2.54−4.40i)T+(−20.5−35.5i)T2 1 + (2.54 - 4.40i)T + (-20.5 - 35.5i)T^{2}
43 1+(3.37−5.83i)T+(−21.5−37.2i)T2 1 + (3.37 - 5.83i)T + (-21.5 - 37.2i)T^{2}
47 1+(5.79+10.0i)T+(−23.5+40.7i)T2 1 + (5.79 + 10.0i)T + (-23.5 + 40.7i)T^{2}
53 1+(2.38+4.13i)T+(−26.5+45.8i)T2 1 + (2.38 + 4.13i)T + (-26.5 + 45.8i)T^{2}
59 1+(−5.08+8.81i)T+(−29.5−51.0i)T2 1 + (-5.08 + 8.81i)T + (-29.5 - 51.0i)T^{2}
61 1+(1.95+3.38i)T+(−30.5+52.8i)T2 1 + (1.95 + 3.38i)T + (-30.5 + 52.8i)T^{2}
67 1+(1.74+3.03i)T+(−33.5+58.0i)T2 1 + (1.74 + 3.03i)T + (-33.5 + 58.0i)T^{2}
71 1+(4.00−6.94i)T+(−35.5−61.4i)T2 1 + (4.00 - 6.94i)T + (-35.5 - 61.4i)T^{2}
73 1+(−6.00+10.3i)T+(−36.5−63.2i)T2 1 + (-6.00 + 10.3i)T + (-36.5 - 63.2i)T^{2}
79 1+(7.92−13.7i)T+(−39.5−68.4i)T2 1 + (7.92 - 13.7i)T + (-39.5 - 68.4i)T^{2}
83 1−9.16T+83T2 1 - 9.16T + 83T^{2}
89 1+(−1.38−2.40i)T+(−44.5+77.0i)T2 1 + (-1.38 - 2.40i)T + (-44.5 + 77.0i)T^{2}
97 1+(1.63−2.84i)T+(−48.5−84.0i)T2 1 + (1.63 - 2.84i)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

−11.19748465375654414603000285676, −10.30462132776299355806407666782, −9.667134692763502848933375543061, −8.217471776137255609839056461355, −7.895892962115988027978585964212, −6.82173873337306118480819631661, −6.33714873002660654737586355796, −5.10366789626396301021249361276, −3.61943796346771066772990824498, −2.89293342460774757850786765877, 0.23782174260916193640025221720, 1.30690624874635620244968035217, 2.66702108469781293803875890824, 4.21974985222137445343471476183, 5.09431234602299442916850083015, 6.26765202523811461502811187931, 7.54228176580241818279209928847, 8.375312145286623064450972841954, 9.084670340499075947888794474646, 9.851699788759877184399724039351

Graph of the ZZ-function along the critical line