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Properties

Label 2-630-105.104-c1-0-11
Degree 22
Conductor 630630
Sign −0.0250+0.999i-0.0250 + 0.999i
Analytic cond. 5.030575.03057
Root an. cond. 2.242892.24289
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-s + 4-s − 2.23i·5-s + (1.58 − 2.12i)7-s − 8-s + 2.23i·10-s + 1.41i·11-s + 3.16·13-s + (−1.58 + 2.12i)14-s + 16-s − 4.47i·17-s − 2.23i·20-s − 1.41i·22-s − 6·23-s − 5.00·25-s − 3.16·26-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.999i·5-s + (0.597 − 0.801i)7-s − 0.353·8-s + 0.707i·10-s + 0.426i·11-s + 0.877·13-s + (−0.422 + 0.566i)14-s + 0.250·16-s − 1.08i·17-s − 0.499i·20-s − 0.301i·22-s − 1.25·23-s − 1.00·25-s − 0.620·26-s + ⋯

Functional equation

Λ(s)=(630s/2 ΓC(s) L(s)=((−0.0250+0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0250 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(630s/2 ΓC(s+1/2) L(s)=((−0.0250+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0250 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 630630    =    2⋅32⋅5⋅72 \cdot 3^{2} \cdot 5 \cdot 7
Sign: −0.0250+0.999i-0.0250 + 0.999i
Analytic conductor: 5.030575.03057
Root analytic conductor: 2.242892.24289
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ630(629,⋅)\chi_{630} (629, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 630, ( :1/2), −0.0250+0.999i)(2,\ 630,\ (\ :1/2),\ -0.0250 + 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.734716−0.753349i0.734716 - 0.753349i
L(12)L(\frac12) ≈\approx 0.734716−0.753349i0.734716 - 0.753349i
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)
bad2 1+T 1 + T
3 1 1
5 1+2.23iT 1 + 2.23iT
7 1+(−1.58+2.12i)T 1 + (-1.58 + 2.12i)T
good11 1−1.41iT−11T2 1 - 1.41iT - 11T^{2}
13 1−3.16T+13T2 1 - 3.16T + 13T^{2}
17 1+4.47iT−17T2 1 + 4.47iT - 17T^{2}
19 1−19T2 1 - 19T^{2}
23 1+6T+23T2 1 + 6T + 23T^{2}
29 1+2.82iT−29T2 1 + 2.82iT - 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1−4.24iT−37T2 1 - 4.24iT - 37T^{2}
41 1−9.48T+41T2 1 - 9.48T + 41T^{2}
43 1+8.48iT−43T2 1 + 8.48iT - 43T^{2}
47 1+4.47iT−47T2 1 + 4.47iT - 47T^{2}
53 1+6T+53T2 1 + 6T + 53T^{2}
59 1−9.48T+59T2 1 - 9.48T + 59T^{2}
61 1+13.4iT−61T2 1 + 13.4iT - 61T^{2}
67 1−67T2 1 - 67T^{2}
71 1−5.65iT−71T2 1 - 5.65iT - 71T^{2}
73 1+6.32T+73T2 1 + 6.32T + 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1−8.94iT−83T2 1 - 8.94iT - 83T^{2}
89 1+9.48T+89T2 1 + 9.48T + 89T^{2}
97 1−12.6T+97T2 1 - 12.6T + 97T^{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

−10.19019245128368008615632834623, −9.531236690328958957651902900364, −8.558511955016878264934298298333, −7.925117932707761620388852170211, −7.06977408034962942604364076625, −5.87469206114132527698504709593, −4.77625879851704816360809095817, −3.81808402610915940952594381156, −2.00033142574844190443925101690, −0.75552635560664395036088032976, 1.69990310217715932319361391860, 2.87145376950628508980656054161, 4.07690455696665579503348347992, 5.84316538225360915818631842009, 6.20168260108870078040702839289, 7.50703467499020019248506410240, 8.231401445990579585561071686273, 8.981022815354623021280794898161, 10.02241672483821613225587595146, 10.86960207532378637024513611208

Graph of the ZZ-function along the critical line