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Properties

Label 2-1470-7.2-c1-0-3
Degree 22
Conductor 14701470
Sign −0.198−0.980i-0.198 - 0.980i
Analytic cond. 11.738011.7380
Root an. cond. 3.426073.42607
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  + (−0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−0.5 + 0.866i)5-s + 0.999·6-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.499 − 0.866i)10-s + (−1.70 − 2.95i)11-s + (−0.499 + 0.866i)12-s + 0.999·15-s + (−0.5 + 0.866i)16-s + (0.707 + 1.22i)17-s + (−0.499 − 0.866i)18-s + (−1.41 + 2.44i)19-s + 0.999·20-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.223 + 0.387i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 − 0.273i)10-s + (−0.514 − 0.891i)11-s + (−0.144 + 0.249i)12-s + 0.258·15-s + (−0.125 + 0.216i)16-s + (0.171 + 0.297i)17-s + (−0.117 − 0.204i)18-s + (−0.324 + 0.561i)19-s + 0.223·20-s + ⋯

Functional equation

Λ(s)=(1470s/2 ΓC(s) L(s)=((−0.198−0.980i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1470s/2 ΓC(s+1/2) L(s)=((−0.198−0.980i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14701470    =    2⋅3⋅5⋅722 \cdot 3 \cdot 5 \cdot 7^{2}
Sign: −0.198−0.980i-0.198 - 0.980i
Analytic conductor: 11.738011.7380
Root analytic conductor: 3.426073.42607
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1470(961,⋅)\chi_{1470} (961, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1470, ( :1/2), −0.198−0.980i)(2,\ 1470,\ (\ :1/2),\ -0.198 - 0.980i)

Particular Values

L(1)L(1) ≈\approx 0.76989535060.7698953506
L(12)L(\frac12) ≈\approx 0.76989535060.7698953506
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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
5 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
7 1 1
good11 1+(1.70+2.95i)T+(−5.5+9.52i)T2 1 + (1.70 + 2.95i)T + (-5.5 + 9.52i)T^{2}
13 1+13T2 1 + 13T^{2}
17 1+(−0.707−1.22i)T+(−8.5+14.7i)T2 1 + (-0.707 - 1.22i)T + (-8.5 + 14.7i)T^{2}
19 1+(1.41−2.44i)T+(−9.5−16.4i)T2 1 + (1.41 - 2.44i)T + (-9.5 - 16.4i)T^{2}
23 1+(−0.414+0.717i)T+(−11.5−19.9i)T2 1 + (-0.414 + 0.717i)T + (-11.5 - 19.9i)T^{2}
29 1+0.242T+29T2 1 + 0.242T + 29T^{2}
31 1+(−4.53−7.85i)T+(−15.5+26.8i)T2 1 + (-4.53 - 7.85i)T + (-15.5 + 26.8i)T^{2}
37 1+(0.707−1.22i)T+(−18.5−32.0i)T2 1 + (0.707 - 1.22i)T + (-18.5 - 32.0i)T^{2}
41 1+3.17T+41T2 1 + 3.17T + 41T^{2}
43 1−7.41T+43T2 1 - 7.41T + 43T^{2}
47 1+(2.53−4.39i)T+(−23.5−40.7i)T2 1 + (2.53 - 4.39i)T + (-23.5 - 40.7i)T^{2}
53 1+(6.65+11.5i)T+(−26.5+45.8i)T2 1 + (6.65 + 11.5i)T + (-26.5 + 45.8i)T^{2}
59 1+(−7.24−12.5i)T+(−29.5+51.0i)T2 1 + (-7.24 - 12.5i)T + (-29.5 + 51.0i)T^{2}
61 1+(−0.171+0.297i)T+(−30.5−52.8i)T2 1 + (-0.171 + 0.297i)T + (-30.5 - 52.8i)T^{2}
67 1+(−5.94−10.3i)T+(−33.5+58.0i)T2 1 + (-5.94 - 10.3i)T + (-33.5 + 58.0i)T^{2}
71 1−5.17T+71T2 1 - 5.17T + 71T^{2}
73 1+(−1.82−3.16i)T+(−36.5+63.2i)T2 1 + (-1.82 - 3.16i)T + (-36.5 + 63.2i)T^{2}
79 1+(5.65−9.79i)T+(−39.5−68.4i)T2 1 + (5.65 - 9.79i)T + (-39.5 - 68.4i)T^{2}
83 1+10.8T+83T2 1 + 10.8T + 83T^{2}
89 1+(5.24−9.08i)T+(−44.5−77.0i)T2 1 + (5.24 - 9.08i)T + (-44.5 - 77.0i)T^{2}
97 1−3.17T+97T2 1 - 3.17T + 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

−9.754810330913411151507361382501, −8.495021314128241255974282419760, −8.246410669436256660356223918726, −7.25989299742626737596903775879, −6.55644702096843847254803467350, −5.82036260820054687766156706558, −5.00993183306606390268070222634, −3.75591419435204632760761275178, −2.59711669259207851235398988256, −1.11112538662217581010700687359, 0.41727357623473776281040838839, 1.99958523418173970202159386959, 3.10427964429370067315110536756, 4.27284539593529618821790422922, 4.81510848255930719762375182890, 5.83863340725434712325054509560, 7.01496126336090822623716916298, 7.82872520794894513050446717953, 8.627921649640137616815804712483, 9.552909842787325594749067748140

Graph of the ZZ-function along the critical line