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Properties

Label 2-637-91.16-c1-0-11
Degree 22
Conductor 637637
Sign 0.355−0.934i0.355 - 0.934i
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  − 0.579·2-s + (−0.946 + 1.63i)3-s − 1.66·4-s + (0.736 − 1.27i)5-s + (0.548 − 0.950i)6-s + 2.12·8-s + (−0.289 − 0.502i)9-s + (−0.427 + 0.739i)10-s + (−0.289 + 0.502i)11-s + (1.57 − 2.72i)12-s + (0.128 − 3.60i)13-s + (1.39 + 2.41i)15-s + 2.09·16-s + 1.19·17-s + (0.168 + 0.291i)18-s + (−0.230 − 0.399i)19-s + ⋯
L(s)  = 1  − 0.409·2-s + (−0.546 + 0.946i)3-s − 0.831·4-s + (0.329 − 0.570i)5-s + (0.223 − 0.387i)6-s + 0.751·8-s + (−0.0966 − 0.167i)9-s + (−0.135 + 0.233i)10-s + (−0.0874 + 0.151i)11-s + (0.454 − 0.787i)12-s + (0.0357 − 0.999i)13-s + (0.359 + 0.623i)15-s + 0.523·16-s + 0.290·17-s + (0.0396 + 0.0686i)18-s + (−0.0528 − 0.0915i)19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((0.355−0.934i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((0.355−0.934i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: 0.355−0.934i0.355 - 0.934i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(471,⋅)\chi_{637} (471, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), 0.355−0.934i)(2,\ 637,\ (\ :1/2),\ 0.355 - 0.934i)

Particular Values

L(1)L(1) ≈\approx 0.673760+0.464427i0.673760 + 0.464427i
L(12)L(\frac12) ≈\approx 0.673760+0.464427i0.673760 + 0.464427i
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+(−0.128+3.60i)T 1 + (-0.128 + 3.60i)T
good2 1+0.579T+2T2 1 + 0.579T + 2T^{2}
3 1+(0.946−1.63i)T+(−1.5−2.59i)T2 1 + (0.946 - 1.63i)T + (-1.5 - 2.59i)T^{2}
5 1+(−0.736+1.27i)T+(−2.5−4.33i)T2 1 + (-0.736 + 1.27i)T + (-2.5 - 4.33i)T^{2}
11 1+(0.289−0.502i)T+(−5.5−9.52i)T2 1 + (0.289 - 0.502i)T + (-5.5 - 9.52i)T^{2}
17 1−1.19T+17T2 1 - 1.19T + 17T^{2}
19 1+(0.230+0.399i)T+(−9.5+16.4i)T2 1 + (0.230 + 0.399i)T + (-9.5 + 16.4i)T^{2}
23 1−2.36T+23T2 1 - 2.36T + 23T^{2}
29 1+(−3.44−5.96i)T+(−14.5+25.1i)T2 1 + (-3.44 - 5.96i)T + (-14.5 + 25.1i)T^{2}
31 1+(−2.22−3.84i)T+(−15.5+26.8i)T2 1 + (-2.22 - 3.84i)T + (-15.5 + 26.8i)T^{2}
37 1−9.16T+37T2 1 - 9.16T + 37T^{2}
41 1+(2.00+3.47i)T+(−20.5+35.5i)T2 1 + (2.00 + 3.47i)T + (-20.5 + 35.5i)T^{2}
43 1+(4.02−6.97i)T+(−21.5−37.2i)T2 1 + (4.02 - 6.97i)T + (-21.5 - 37.2i)T^{2}
47 1+(5.75−9.97i)T+(−23.5−40.7i)T2 1 + (5.75 - 9.97i)T + (-23.5 - 40.7i)T^{2}
53 1+(−4.69−8.13i)T+(−26.5+45.8i)T2 1 + (-4.69 - 8.13i)T + (-26.5 + 45.8i)T^{2}
59 1+0.240T+59T2 1 + 0.240T + 59T^{2}
61 1+(−3.86−6.69i)T+(−30.5+52.8i)T2 1 + (-3.86 - 6.69i)T + (-30.5 + 52.8i)T^{2}
67 1+(−0.724+1.25i)T+(−33.5−58.0i)T2 1 + (-0.724 + 1.25i)T + (-33.5 - 58.0i)T^{2}
71 1+(−6.25+10.8i)T+(−35.5−61.4i)T2 1 + (-6.25 + 10.8i)T + (-35.5 - 61.4i)T^{2}
73 1+(1.84+3.19i)T+(−36.5+63.2i)T2 1 + (1.84 + 3.19i)T + (-36.5 + 63.2i)T^{2}
79 1+(8.03−13.9i)T+(−39.5−68.4i)T2 1 + (8.03 - 13.9i)T + (-39.5 - 68.4i)T^{2}
83 1−15.4T+83T2 1 - 15.4T + 83T^{2}
89 1+2.49T+89T2 1 + 2.49T + 89T^{2}
97 1+(−7.82+13.5i)T+(−48.5−84.0i)T2 1 + (-7.82 + 13.5i)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

−10.49550404578259339978565006478, −9.896128757694921097189792361247, −9.184507679584010355832707849004, −8.371983832712743482142336303237, −7.44530904655738560712282027159, −5.91076266754457141825335745633, −5.01187757739884982110026878344, −4.59843725726093570780371400235, −3.24683351912744735327667771579, −1.11492703540708945793058665478, 0.72273975159116333587483893882, 2.14701931989856412948559058917, 3.81681242304431167174811833702, 4.98287107339678412136637646125, 6.16782541660733044621504550857, 6.78770393394062905603242249766, 7.77347675637450382920699975295, 8.609825280858296438312486997968, 9.669204887449777328942757721623, 10.21486810813399703292187015339

Graph of the ZZ-function along the critical line