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Properties

Label 2-630-63.25-c1-0-30
Degree 22
Conductor 630630
Sign −0.990+0.139i-0.990 + 0.139i
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 + (0.0520 − 1.73i)3-s + 4-s + (0.5 − 0.866i)5-s + (−0.0520 + 1.73i)6-s + (0.226 − 2.63i)7-s − 8-s + (−2.99 − 0.180i)9-s + (−0.5 + 0.866i)10-s + (−2.57 − 4.45i)11-s + (0.0520 − 1.73i)12-s + (1.62 + 2.81i)13-s + (−0.226 + 2.63i)14-s + (−1.47 − 0.910i)15-s + 16-s + (−2.89 + 5.01i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.0300 − 0.999i)3-s + 0.5·4-s + (0.223 − 0.387i)5-s + (−0.0212 + 0.706i)6-s + (0.0856 − 0.996i)7-s − 0.353·8-s + (−0.998 − 0.0601i)9-s + (−0.158 + 0.273i)10-s + (−0.775 − 1.34i)11-s + (0.0150 − 0.499i)12-s + (0.450 + 0.779i)13-s + (−0.0605 + 0.704i)14-s + (−0.380 − 0.235i)15-s + 0.250·16-s + (−0.702 + 1.21i)17-s + ⋯

Functional equation

Λ(s)=(630s/2 ΓC(s) L(s)=((−0.990+0.139i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(630s/2 ΓC(s+1/2) L(s)=((−0.990+0.139i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 630630    =    2⋅32⋅5⋅72 \cdot 3^{2} \cdot 5 \cdot 7
Sign: −0.990+0.139i-0.990 + 0.139i
Analytic conductor: 5.030575.03057
Root analytic conductor: 2.242892.24289
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ630(151,⋅)\chi_{630} (151, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 630, ( :1/2), −0.990+0.139i)(2,\ 630,\ (\ :1/2),\ -0.990 + 0.139i)

Particular Values

L(1)L(1) ≈\approx 0.0520313−0.743021i0.0520313 - 0.743021i
L(12)L(\frac12) ≈\approx 0.0520313−0.743021i0.0520313 - 0.743021i
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+(−0.0520+1.73i)T 1 + (-0.0520 + 1.73i)T
5 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
7 1+(−0.226+2.63i)T 1 + (-0.226 + 2.63i)T
good11 1+(2.57+4.45i)T+(−5.5+9.52i)T2 1 + (2.57 + 4.45i)T + (-5.5 + 9.52i)T^{2}
13 1+(−1.62−2.81i)T+(−6.5+11.2i)T2 1 + (-1.62 - 2.81i)T + (-6.5 + 11.2i)T^{2}
17 1+(2.89−5.01i)T+(−8.5−14.7i)T2 1 + (2.89 - 5.01i)T + (-8.5 - 14.7i)T^{2}
19 1+(2.34+4.06i)T+(−9.5+16.4i)T2 1 + (2.34 + 4.06i)T + (-9.5 + 16.4i)T^{2}
23 1+(−1.33+2.31i)T+(−11.5−19.9i)T2 1 + (-1.33 + 2.31i)T + (-11.5 - 19.9i)T^{2}
29 1+(−0.227+0.394i)T+(−14.5−25.1i)T2 1 + (-0.227 + 0.394i)T + (-14.5 - 25.1i)T^{2}
31 1−7.14T+31T2 1 - 7.14T + 31T^{2}
37 1+(−2.00−3.46i)T+(−18.5+32.0i)T2 1 + (-2.00 - 3.46i)T + (-18.5 + 32.0i)T^{2}
41 1+(3.71+6.42i)T+(−20.5+35.5i)T2 1 + (3.71 + 6.42i)T + (-20.5 + 35.5i)T^{2}
43 1+(1.80−3.12i)T+(−21.5−37.2i)T2 1 + (1.80 - 3.12i)T + (-21.5 - 37.2i)T^{2}
47 1+3.15T+47T2 1 + 3.15T + 47T^{2}
53 1+(1.63−2.82i)T+(−26.5−45.8i)T2 1 + (1.63 - 2.82i)T + (-26.5 - 45.8i)T^{2}
59 1+11.1T+59T2 1 + 11.1T + 59T^{2}
61 1−5.99T+61T2 1 - 5.99T + 61T^{2}
67 1+11.7T+67T2 1 + 11.7T + 67T^{2}
71 1+4.38T+71T2 1 + 4.38T + 71T^{2}
73 1+(−6.94+12.0i)T+(−36.5−63.2i)T2 1 + (-6.94 + 12.0i)T + (-36.5 - 63.2i)T^{2}
79 1−2.95T+79T2 1 - 2.95T + 79T^{2}
83 1+(−6.13+10.6i)T+(−41.5−71.8i)T2 1 + (-6.13 + 10.6i)T + (-41.5 - 71.8i)T^{2}
89 1+(4.83+8.36i)T+(−44.5+77.0i)T2 1 + (4.83 + 8.36i)T + (-44.5 + 77.0i)T^{2}
97 1+(−3.91+6.78i)T+(−48.5−84.0i)T2 1 + (-3.91 + 6.78i)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

−10.37782403566703867192471437547, −8.928575420481612075140675955261, −8.474463934309499359584352340524, −7.69943779808146862889903482360, −6.57168048470446678511042706421, −6.13301856755540501086194312419, −4.60851409487746359945791476466, −3.10168840477517416971764563601, −1.74492879988632525760616594683, −0.48967435316127325420660519437, 2.22940446256406910489669197778, 3.09551448178640134109120523131, 4.67259988780745749858699721357, 5.51955034829734599438611257499, 6.52968911226036577755446030773, 7.77131475701518352519224205686, 8.520286283305729574271514064106, 9.497355588483840981358965791470, 9.979422486370781059093671693161, 10.76888515532100329050403365301

Graph of the ZZ-function along the critical line