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Properties

Label 2-637-91.81-c1-0-2
Degree 22
Conductor 637637
Sign −0.270−0.962i-0.270 - 0.962i
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  + (0.651 − 1.12i)2-s − 2.88·3-s + (0.151 + 0.262i)4-s + (1.44 + 2.49i)5-s + (−1.87 + 3.25i)6-s + 3·8-s + 5.30·9-s + 3.75·10-s − 5.90·11-s + (−0.436 − 0.755i)12-s + (−3.31 + 1.41i)13-s + (−4.15 − 7.19i)15-s + (1.65 − 2.86i)16-s + (−0.436 − 0.755i)17-s + (3.45 − 5.98i)18-s − 2.88·19-s + ⋯
L(s)  = 1  + (0.460 − 0.797i)2-s − 1.66·3-s + (0.0756 + 0.131i)4-s + (0.644 + 1.11i)5-s + (−0.766 + 1.32i)6-s + 1.06·8-s + 1.76·9-s + 1.18·10-s − 1.78·11-s + (−0.125 − 0.218i)12-s + (−0.920 + 0.391i)13-s + (−1.07 − 1.85i)15-s + (0.412 − 0.715i)16-s + (−0.105 − 0.183i)17-s + (0.814 − 1.41i)18-s − 0.661·19-s + ⋯

Functional equation

Λ(s)=(637s/2 ΓC(s) L(s)=((−0.270−0.962i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(637s/2 ΓC(s+1/2) L(s)=((−0.270−0.962i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 637637    =    72⋅137^{2} \cdot 13
Sign: −0.270−0.962i-0.270 - 0.962i
Analytic conductor: 5.086475.08647
Root analytic conductor: 2.255322.25532
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ637(263,⋅)\chi_{637} (263, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 637, ( :1/2), −0.270−0.962i)(2,\ 637,\ (\ :1/2),\ -0.270 - 0.962i)

Particular Values

L(1)L(1) ≈\approx 0.401493+0.529721i0.401493 + 0.529721i
L(12)L(\frac12) ≈\approx 0.401493+0.529721i0.401493 + 0.529721i
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+(3.31−1.41i)T 1 + (3.31 - 1.41i)T
good2 1+(−0.651+1.12i)T+(−1−1.73i)T2 1 + (-0.651 + 1.12i)T + (-1 - 1.73i)T^{2}
3 1+2.88T+3T2 1 + 2.88T + 3T^{2}
5 1+(−1.44−2.49i)T+(−2.5+4.33i)T2 1 + (-1.44 - 2.49i)T + (-2.5 + 4.33i)T^{2}
11 1+5.90T+11T2 1 + 5.90T + 11T^{2}
17 1+(0.436+0.755i)T+(−8.5+14.7i)T2 1 + (0.436 + 0.755i)T + (-8.5 + 14.7i)T^{2}
19 1+2.88T+19T2 1 + 2.88T + 19T^{2}
23 1+(3.30−5.72i)T+(−11.5−19.9i)T2 1 + (3.30 - 5.72i)T + (-11.5 - 19.9i)T^{2}
29 1+(0.651+1.12i)T+(−14.5+25.1i)T2 1 + (0.651 + 1.12i)T + (-14.5 + 25.1i)T^{2}
31 1+(−0.436+0.755i)T+(−15.5−26.8i)T2 1 + (-0.436 + 0.755i)T + (-15.5 - 26.8i)T^{2}
37 1+(0.697−1.20i)T+(−18.5−32.0i)T2 1 + (0.697 - 1.20i)T + (-18.5 - 32.0i)T^{2}
41 1+(−3.75−6.50i)T+(−20.5+35.5i)T2 1 + (-3.75 - 6.50i)T + (-20.5 + 35.5i)T^{2}
43 1+(2.75−4.77i)T+(−21.5−37.2i)T2 1 + (2.75 - 4.77i)T + (-21.5 - 37.2i)T^{2}
47 1+(6.19+10.7i)T+(−23.5+40.7i)T2 1 + (6.19 + 10.7i)T + (-23.5 + 40.7i)T^{2}
53 1+(4.80−8.31i)T+(−26.5−45.8i)T2 1 + (4.80 - 8.31i)T + (-26.5 - 45.8i)T^{2}
59 1+(−3.31−5.74i)T+(−29.5+51.0i)T2 1 + (-3.31 - 5.74i)T + (-29.5 + 51.0i)T^{2}
61 1+5.76T+61T2 1 + 5.76T + 61T^{2}
67 1−T+67T2 1 - T + 67T^{2}
71 1+(2−3.46i)T+(−35.5−61.4i)T2 1 + (2 - 3.46i)T + (-35.5 - 61.4i)T^{2}
73 1+(2.88−4.99i)T+(−36.5−63.2i)T2 1 + (2.88 - 4.99i)T + (-36.5 - 63.2i)T^{2}
79 1+(0.302+0.524i)T+(−39.5+68.4i)T2 1 + (0.302 + 0.524i)T + (-39.5 + 68.4i)T^{2}
83 1−6.63T+83T2 1 - 6.63T + 83T^{2}
89 1+(−4.32+7.48i)T+(−44.5−77.0i)T2 1 + (-4.32 + 7.48i)T + (-44.5 - 77.0i)T^{2}
97 1+(3.88−6.73i)T+(−48.5−84.0i)T2 1 + (3.88 - 6.73i)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

−10.98561458214860426518022685502, −10.24579739677536565264119769932, −9.912465429173331996474945934118, −7.82688317178886329083062210592, −7.11530423119909990739003302320, −6.19711589203445344710549032243, −5.29186700700751361976984848614, −4.48289715320759449117064495917, −2.96290337409701181636073737890, −2.00406772008254332923907426509, 0.35232875820824479073730201156, 2.02734045614282832183783224516, 4.65434739539467427610017102900, 4.99016298812799822771600940533, 5.68690271964038464712687686915, 6.34428015578737401457285997680, 7.39689825945529928656404350941, 8.309639699189783011609097856314, 9.794535540601185319282320462248, 10.46473694669118604717579500192

Graph of the ZZ-function along the critical line