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Properties

Label 2-627-11.3-c1-0-23
Degree 22
Conductor 627627
Sign 0.999+0.00420i0.999 + 0.00420i
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  + (1.74 − 1.26i)2-s + (0.309 + 0.951i)3-s + (0.815 − 2.51i)4-s + (2.24 + 1.63i)5-s + (1.74 + 1.26i)6-s + (−0.940 + 2.89i)7-s + (−0.425 − 1.30i)8-s + (−0.809 + 0.587i)9-s + 5.98·10-s + (0.962 − 3.17i)11-s + 2.63·12-s + (−2.31 + 1.67i)13-s + (2.02 + 6.23i)14-s + (−0.858 + 2.64i)15-s + (1.87 + 1.35i)16-s + (−2.15 − 1.56i)17-s + ⋯
L(s)  = 1  + (1.23 − 0.895i)2-s + (0.178 + 0.549i)3-s + (0.407 − 1.25i)4-s + (1.00 + 0.730i)5-s + (0.711 + 0.516i)6-s + (−0.355 + 1.09i)7-s + (−0.150 − 0.462i)8-s + (−0.269 + 0.195i)9-s + 1.89·10-s + (0.290 − 0.956i)11-s + 0.761·12-s + (−0.641 + 0.465i)13-s + (0.541 + 1.66i)14-s + (−0.221 + 0.682i)15-s + (0.467 + 0.339i)16-s + (−0.523 − 0.380i)17-s + ⋯

Functional equation

Λ(s)=(627s/2 ΓC(s) L(s)=((0.999+0.00420i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00420i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(627s/2 ΓC(s+1/2) L(s)=((0.999+0.00420i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00420i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 627627    =    3⋅11⋅193 \cdot 11 \cdot 19
Sign: 0.999+0.00420i0.999 + 0.00420i
Analytic conductor: 5.006625.00662
Root analytic conductor: 2.237542.23754
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ627(58,⋅)\chi_{627} (58, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 627, ( :1/2), 0.999+0.00420i)(2,\ 627,\ (\ :1/2),\ 0.999 + 0.00420i)

Particular Values

L(1)L(1) ≈\approx 3.28892−0.00691895i3.28892 - 0.00691895i
L(12)L(\frac12) ≈\approx 3.28892−0.00691895i3.28892 - 0.00691895i
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+(−0.309−0.951i)T 1 + (-0.309 - 0.951i)T
11 1+(−0.962+3.17i)T 1 + (-0.962 + 3.17i)T
19 1+(−0.309−0.951i)T 1 + (-0.309 - 0.951i)T
good2 1+(−1.74+1.26i)T+(0.618−1.90i)T2 1 + (-1.74 + 1.26i)T + (0.618 - 1.90i)T^{2}
5 1+(−2.24−1.63i)T+(1.54+4.75i)T2 1 + (-2.24 - 1.63i)T + (1.54 + 4.75i)T^{2}
7 1+(0.940−2.89i)T+(−5.66−4.11i)T2 1 + (0.940 - 2.89i)T + (-5.66 - 4.11i)T^{2}
13 1+(2.31−1.67i)T+(4.01−12.3i)T2 1 + (2.31 - 1.67i)T + (4.01 - 12.3i)T^{2}
17 1+(2.15+1.56i)T+(5.25+16.1i)T2 1 + (2.15 + 1.56i)T + (5.25 + 16.1i)T^{2}
23 1+2.93T+23T2 1 + 2.93T + 23T^{2}
29 1+(−2.73+8.42i)T+(−23.4−17.0i)T2 1 + (-2.73 + 8.42i)T + (-23.4 - 17.0i)T^{2}
31 1+(−8.39+6.09i)T+(9.57−29.4i)T2 1 + (-8.39 + 6.09i)T + (9.57 - 29.4i)T^{2}
37 1+(0.201−0.621i)T+(−29.9−21.7i)T2 1 + (0.201 - 0.621i)T + (-29.9 - 21.7i)T^{2}
41 1+(2.52+7.78i)T+(−33.1+24.0i)T2 1 + (2.52 + 7.78i)T + (-33.1 + 24.0i)T^{2}
43 1+1.85T+43T2 1 + 1.85T + 43T^{2}
47 1+(−0.0728−0.224i)T+(−38.0+27.6i)T2 1 + (-0.0728 - 0.224i)T + (-38.0 + 27.6i)T^{2}
53 1+(1.93−1.40i)T+(16.3−50.4i)T2 1 + (1.93 - 1.40i)T + (16.3 - 50.4i)T^{2}
59 1+(−0.795+2.44i)T+(−47.7−34.6i)T2 1 + (-0.795 + 2.44i)T + (-47.7 - 34.6i)T^{2}
61 1+(7.22+5.25i)T+(18.8+58.0i)T2 1 + (7.22 + 5.25i)T + (18.8 + 58.0i)T^{2}
67 1+2.05T+67T2 1 + 2.05T + 67T^{2}
71 1+(3.74+2.72i)T+(21.9+67.5i)T2 1 + (3.74 + 2.72i)T + (21.9 + 67.5i)T^{2}
73 1+(1.02−3.16i)T+(−59.0−42.9i)T2 1 + (1.02 - 3.16i)T + (-59.0 - 42.9i)T^{2}
79 1+(8.83−6.42i)T+(24.4−75.1i)T2 1 + (8.83 - 6.42i)T + (24.4 - 75.1i)T^{2}
83 1+(−14.5−10.5i)T+(25.6+78.9i)T2 1 + (-14.5 - 10.5i)T + (25.6 + 78.9i)T^{2}
89 1−7.08T+89T2 1 - 7.08T + 89T^{2}
97 1+(−0.559+0.406i)T+(29.9−92.2i)T2 1 + (-0.559 + 0.406i)T + (29.9 - 92.2i)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

−10.69362867878529367308180425293, −9.936352117185017180098338115916, −9.266462210257740234085157046241, −8.148452662356928908024549773359, −6.33367904138413958422803504627, −5.97026841821169584695860226403, −4.94200732217966189916115490734, −3.85352617606113868172911663670, −2.67176620687367197479850348063, −2.28869101492042807502503311655, 1.43037219315938329934581910749, 3.07188823499603041648821718627, 4.43392082079960876145752504282, 5.02788637576789611220590079419, 6.17010485295901101487300800194, 6.83710805283468452390938076365, 7.52273873835406857029024069148, 8.651848115478939008919826406440, 9.808252545897061492540757248343, 10.38589828303980747014883459873

Graph of the ZZ-function along the critical line