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Properties

Label 2-315-105.2-c1-0-3
Degree 22
Conductor 315315
Sign 0.650+0.759i0.650 + 0.759i
Analytic cond. 2.515282.51528
Root an. cond. 1.585961.58596
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  + (−0.695 − 2.59i)2-s + (−4.52 + 2.61i)4-s + (1.27 + 1.83i)5-s + (−1.50 + 2.17i)7-s + (6.14 + 6.14i)8-s + (3.88 − 4.58i)10-s + (3.60 − 2.07i)11-s + (−0.702 + 0.702i)13-s + (6.69 + 2.40i)14-s + (6.44 − 11.1i)16-s + (1.31 + 0.352i)17-s + (5.22 + 3.01i)19-s + (−10.5 − 5.00i)20-s + (−7.90 − 7.90i)22-s + (0.205 − 0.0550i)23-s + ⋯
L(s)  = 1  + (−0.492 − 1.83i)2-s + (−2.26 + 1.30i)4-s + (0.569 + 0.822i)5-s + (−0.569 + 0.821i)7-s + (2.17 + 2.17i)8-s + (1.23 − 1.44i)10-s + (1.08 − 0.626i)11-s + (−0.194 + 0.194i)13-s + (1.78 + 0.641i)14-s + (1.61 − 2.79i)16-s + (0.319 + 0.0855i)17-s + (1.19 + 0.692i)19-s + (−2.36 − 1.11i)20-s + (−1.68 − 1.68i)22-s + (0.0428 − 0.0114i)23-s + ⋯

Functional equation

Λ(s)=(315s/2 ΓC(s) L(s)=((0.650+0.759i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(315s/2 ΓC(s+1/2) L(s)=((0.650+0.759i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 315315    =    32⋅5⋅73^{2} \cdot 5 \cdot 7
Sign: 0.650+0.759i0.650 + 0.759i
Analytic conductor: 2.515282.51528
Root analytic conductor: 1.585961.58596
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ315(107,⋅)\chi_{315} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 315, ( :1/2), 0.650+0.759i)(2,\ 315,\ (\ :1/2),\ 0.650 + 0.759i)

Particular Values

L(1)L(1) ≈\approx 0.852045−0.392155i0.852045 - 0.392155i
L(12)L(\frac12) ≈\approx 0.852045−0.392155i0.852045 - 0.392155i
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 1
5 1+(−1.27−1.83i)T 1 + (-1.27 - 1.83i)T
7 1+(1.50−2.17i)T 1 + (1.50 - 2.17i)T
good2 1+(0.695+2.59i)T+(−1.73+i)T2 1 + (0.695 + 2.59i)T + (-1.73 + i)T^{2}
11 1+(−3.60+2.07i)T+(5.5−9.52i)T2 1 + (-3.60 + 2.07i)T + (5.5 - 9.52i)T^{2}
13 1+(0.702−0.702i)T−13iT2 1 + (0.702 - 0.702i)T - 13iT^{2}
17 1+(−1.31−0.352i)T+(14.7+8.5i)T2 1 + (-1.31 - 0.352i)T + (14.7 + 8.5i)T^{2}
19 1+(−5.22−3.01i)T+(9.5+16.4i)T2 1 + (-5.22 - 3.01i)T + (9.5 + 16.4i)T^{2}
23 1+(−0.205+0.0550i)T+(19.9−11.5i)T2 1 + (-0.205 + 0.0550i)T + (19.9 - 11.5i)T^{2}
29 1+0.879T+29T2 1 + 0.879T + 29T^{2}
31 1+(−4.76−8.25i)T+(−15.5+26.8i)T2 1 + (-4.76 - 8.25i)T + (-15.5 + 26.8i)T^{2}
37 1+(3.13−0.840i)T+(32.0−18.5i)T2 1 + (3.13 - 0.840i)T + (32.0 - 18.5i)T^{2}
41 1+1.25iT−41T2 1 + 1.25iT - 41T^{2}
43 1+(4.28−4.28i)T−43iT2 1 + (4.28 - 4.28i)T - 43iT^{2}
47 1+(−0.0279−0.104i)T+(−40.7+23.5i)T2 1 + (-0.0279 - 0.104i)T + (-40.7 + 23.5i)T^{2}
53 1+(−1.29+4.83i)T+(−45.8−26.5i)T2 1 + (-1.29 + 4.83i)T + (-45.8 - 26.5i)T^{2}
59 1+(0.113+0.195i)T+(−29.5+51.0i)T2 1 + (0.113 + 0.195i)T + (-29.5 + 51.0i)T^{2}
61 1+(−5.96+10.3i)T+(−30.5−52.8i)T2 1 + (-5.96 + 10.3i)T + (-30.5 - 52.8i)T^{2}
67 1+(−0.825+3.07i)T+(−58.0−33.5i)T2 1 + (-0.825 + 3.07i)T + (-58.0 - 33.5i)T^{2}
71 1−9.78iT−71T2 1 - 9.78iT - 71T^{2}
73 1+(6.60+1.77i)T+(63.2+36.5i)T2 1 + (6.60 + 1.77i)T + (63.2 + 36.5i)T^{2}
79 1+(−1.88−1.08i)T+(39.5+68.4i)T2 1 + (-1.88 - 1.08i)T + (39.5 + 68.4i)T^{2}
83 1+(−3.75−3.75i)T+83iT2 1 + (-3.75 - 3.75i)T + 83iT^{2}
89 1+(−7.19+12.4i)T+(−44.5−77.0i)T2 1 + (-7.19 + 12.4i)T + (-44.5 - 77.0i)T^{2}
97 1+(−3.88−3.88i)T+97iT2 1 + (-3.88 - 3.88i)T + 97iT^{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

−11.64925182294519774996742178653, −10.56397718179861222124608980444, −9.777532495478175120781824855725, −9.219432240801524212603631958176, −8.237462818600208122361085952590, −6.66405042596414151483771232001, −5.33398666731274032329372758380, −3.59991498204569265807009744707, −2.93374556248231806305337119372, −1.56560471352219215960567494840, 0.934433974218193797945875637973, 4.10913912423200682631138322572, 5.08632895095060095937911089290, 6.11433201013670993709758497956, 6.99311247266673212468857141002, 7.76537417741093139604073121272, 8.956688489385521609811856772929, 9.570655322053557091643685044327, 10.16914540084063677208953730609, 11.97720194985623203652851293476

Graph of the ZZ-function along the critical line