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Properties

Label 2-637-637.543-c1-0-41
Degree 22
Conductor 637637
Sign −0.629+0.776i-0.629 + 0.776i
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.36 − 0.177i)2-s + (0.532 − 2.33i)3-s + (3.58 + 0.540i)4-s + (0.329 + 1.06i)5-s + (−1.67 + 5.42i)6-s + (1.40 − 2.24i)7-s + (−3.76 − 0.859i)8-s + (−2.45 − 1.18i)9-s + (−0.590 − 2.58i)10-s + (1.47 + 3.05i)11-s + (3.17 − 8.08i)12-s + (−3.11 − 1.82i)13-s + (−3.72 + 5.05i)14-s + (2.66 − 0.199i)15-s + (1.82 + 0.562i)16-s + (0.844 − 2.15i)17-s + ⋯
L(s)  = 1  + (−1.67 − 0.125i)2-s + (0.307 − 1.34i)3-s + (1.79 + 0.270i)4-s + (0.147 + 0.477i)5-s + (−0.683 + 2.21i)6-s + (0.532 − 0.846i)7-s + (−1.33 − 0.303i)8-s + (−0.818 − 0.394i)9-s + (−0.186 − 0.817i)10-s + (0.443 + 0.921i)11-s + (0.915 − 2.33i)12-s + (−0.863 − 0.505i)13-s + (−0.996 + 1.34i)14-s + (0.688 − 0.0516i)15-s + (0.456 + 0.140i)16-s + (0.204 − 0.521i)17-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.629+0.776i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.629+0.776i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.629 + 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.629+0.776i-0.629 + 0.776i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(543,⋅)\chi_{637} (543, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), −0.629+0.776i)(2,\ 637,\ (\ :1/2),\ -0.629 + 0.776i)

Particular Values

L(1)L(1) ≈\approx 0.317000−0.665066i0.317000 - 0.665066i
L(12)L(\frac12) ≈\approx 0.317000−0.665066i0.317000 - 0.665066i
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+(−1.40+2.24i)T 1 + (-1.40 + 2.24i)T
13 1+(3.11+1.82i)T 1 + (3.11 + 1.82i)T
good2 1+(2.36+0.177i)T+(1.97+0.298i)T2 1 + (2.36 + 0.177i)T + (1.97 + 0.298i)T^{2}
3 1+(−0.532+2.33i)T+(−2.70−1.30i)T2 1 + (-0.532 + 2.33i)T + (-2.70 - 1.30i)T^{2}
5 1+(−0.329−1.06i)T+(−4.13+2.81i)T2 1 + (-0.329 - 1.06i)T + (-4.13 + 2.81i)T^{2}
11 1+(−1.47−3.05i)T+(−6.85+8.60i)T2 1 + (-1.47 - 3.05i)T + (-6.85 + 8.60i)T^{2}
17 1+(−0.844+2.15i)T+(−12.4−11.5i)T2 1 + (-0.844 + 2.15i)T + (-12.4 - 11.5i)T^{2}
19 1+1.22iT−19T2 1 + 1.22iT - 19T^{2}
23 1+(1.36+3.48i)T+(−16.8+15.6i)T2 1 + (1.36 + 3.48i)T + (-16.8 + 15.6i)T^{2}
29 1+(−0.429+1.09i)T+(−21.2−19.7i)T2 1 + (-0.429 + 1.09i)T + (-21.2 - 19.7i)T^{2}
31 1+(−7.21−4.16i)T+(15.5+26.8i)T2 1 + (-7.21 - 4.16i)T + (15.5 + 26.8i)T^{2}
37 1+(1.06+7.06i)T+(−35.3+10.9i)T2 1 + (1.06 + 7.06i)T + (-35.3 + 10.9i)T^{2}
41 1+(0.0827+0.268i)T+(−33.8+23.0i)T2 1 + (0.0827 + 0.268i)T + (-33.8 + 23.0i)T^{2}
43 1+(8.44+2.60i)T+(35.5+24.2i)T2 1 + (8.44 + 2.60i)T + (35.5 + 24.2i)T^{2}
47 1+(0.702−1.02i)T+(−17.1−43.7i)T2 1 + (0.702 - 1.02i)T + (-17.1 - 43.7i)T^{2}
53 1+(−2.12−5.41i)T+(−38.8+36.0i)T2 1 + (-2.12 - 5.41i)T + (-38.8 + 36.0i)T^{2}
59 1+(0.332+0.358i)T+(−4.40+58.8i)T2 1 + (0.332 + 0.358i)T + (-4.40 + 58.8i)T^{2}
61 1+(4.04+5.07i)T+(−13.5+59.4i)T2 1 + (4.04 + 5.07i)T + (-13.5 + 59.4i)T^{2}
67 1+13.3iT−67T2 1 + 13.3iT - 67T^{2}
71 1+(2.63−1.03i)T+(52.0−48.2i)T2 1 + (2.63 - 1.03i)T + (52.0 - 48.2i)T^{2}
73 1+(0.772+1.13i)T+(−26.6+67.9i)T2 1 + (0.772 + 1.13i)T + (-26.6 + 67.9i)T^{2}
79 1+(−4.37−7.58i)T+(−39.5+68.4i)T2 1 + (-4.37 - 7.58i)T + (-39.5 + 68.4i)T^{2}
83 1+(1.46−3.04i)T+(−51.7−64.8i)T2 1 + (1.46 - 3.04i)T + (-51.7 - 64.8i)T^{2}
89 1+(−5.04−7.40i)T+(−32.5+82.8i)T2 1 + (-5.04 - 7.40i)T + (-32.5 + 82.8i)T^{2}
97 1+(−8.94−5.16i)T+(48.5+84.0i)T2 1 + (-8.94 - 5.16i)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

−10.23354059292523284727489304834, −9.389070723627623247983864658531, −8.273326298096821843879340672055, −7.74273293334553246105047405418, −6.95956521605503465620220739962, −6.65880204497924090979010215753, −4.71769437436674798827618043113, −2.74289623046842130622557680598, −1.85582580807134000809605144233, −0.70604409407717312612563176703, 1.52525267029622145298889439762, 2.98300475944543503919572212262, 4.43409548037274909723867864672, 5.45530913006850842205955554062, 6.60098926319274173777663432784, 7.965164023957231740917190207174, 8.584301342189747269036790222662, 9.152327109679487663538100222801, 9.815739641671001852163111946613, 10.42969863029522526211006067389

Graph of the ZZ-function along the critical line