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Properties

Label 2-315-21.20-c9-0-6
Degree 22
Conductor 315315
Sign −0.563−0.825i-0.563 - 0.825i
Analytic cond. 162.236162.236
Root an. cond. 12.737212.7372
Motivic weight 99
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.2i·2-s + 407.·4-s + 625·5-s + (−6.35e3 − 103. i)7-s − 9.38e3i·8-s − 6.37e3i·10-s + 3.22e4i·11-s + 8.60e4i·13-s + (−1.06e3 + 6.48e4i)14-s + 1.12e5·16-s + 1.43e4·17-s − 1.89e5i·19-s + 2.54e5·20-s + 3.29e5·22-s − 5.82e5i·23-s + ⋯
L(s)  = 1  − 0.451i·2-s + 0.796·4-s + 0.447·5-s + (−0.999 − 0.0163i)7-s − 0.810i·8-s − 0.201i·10-s + 0.665i·11-s + 0.835i·13-s + (−0.00737 + 0.451i)14-s + 0.430·16-s + 0.0415·17-s − 0.334i·19-s + 0.356·20-s + 0.300·22-s − 0.433i·23-s + ⋯

Functional equation

Λ(s)=(315s/2 ΓC(s) L(s)=((−0.563−0.825i) Λ‾(10−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.825i)\, \overline{\Lambda}(10-s) \end{aligned}
Λ(s)=(315s/2 ΓC(s+9/2) L(s)=((−0.563−0.825i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.563 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 315315    =    32⋅5⋅73^{2} \cdot 5 \cdot 7
Sign: −0.563−0.825i-0.563 - 0.825i
Analytic conductor: 162.236162.236
Root analytic conductor: 12.737212.7372
Motivic weight: 99
Rational: no
Arithmetic: yes
Character: χ315(251,⋅)\chi_{315} (251, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 315, ( :9/2), −0.563−0.825i)(2,\ 315,\ (\ :9/2),\ -0.563 - 0.825i)

Particular Values

L(5)L(5) ≈\approx 0.63300372030.6330037203
L(12)L(\frac12) ≈\approx 0.63300372030.6330037203
L(112)L(\frac{11}{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 1−625T 1 - 625T
7 1+(6.35e3+103.i)T 1 + (6.35e3 + 103. i)T
good2 1+10.2iT−512T2 1 + 10.2iT - 512T^{2}
11 1−3.22e4iT−2.35e9T2 1 - 3.22e4iT - 2.35e9T^{2}
13 1−8.60e4iT−1.06e10T2 1 - 8.60e4iT - 1.06e10T^{2}
17 1−1.43e4T+1.18e11T2 1 - 1.43e4T + 1.18e11T^{2}
19 1+1.89e5iT−3.22e11T2 1 + 1.89e5iT - 3.22e11T^{2}
23 1+5.82e5iT−1.80e12T2 1 + 5.82e5iT - 1.80e12T^{2}
29 1−5.36e6iT−1.45e13T2 1 - 5.36e6iT - 1.45e13T^{2}
31 1+7.52e6iT−2.64e13T2 1 + 7.52e6iT - 2.64e13T^{2}
37 1−1.95e6T+1.29e14T2 1 - 1.95e6T + 1.29e14T^{2}
41 1+2.23e7T+3.27e14T2 1 + 2.23e7T + 3.27e14T^{2}
43 1+3.30e7T+5.02e14T2 1 + 3.30e7T + 5.02e14T^{2}
47 1−3.49e7T+1.11e15T2 1 - 3.49e7T + 1.11e15T^{2}
53 1−3.02e6iT−3.29e15T2 1 - 3.02e6iT - 3.29e15T^{2}
59 1+6.58e7T+8.66e15T2 1 + 6.58e7T + 8.66e15T^{2}
61 1−1.83e8iT−1.16e16T2 1 - 1.83e8iT - 1.16e16T^{2}
67 1+1.44e8T+2.72e16T2 1 + 1.44e8T + 2.72e16T^{2}
71 1+8.85e7iT−4.58e16T2 1 + 8.85e7iT - 4.58e16T^{2}
73 1−2.24e8iT−5.88e16T2 1 - 2.24e8iT - 5.88e16T^{2}
79 1+8.97e7T+1.19e17T2 1 + 8.97e7T + 1.19e17T^{2}
83 1+4.61e8T+1.86e17T2 1 + 4.61e8T + 1.86e17T^{2}
89 1+1.09e9T+3.50e17T2 1 + 1.09e9T + 3.50e17T^{2}
97 1+2.98e7iT−7.60e17T2 1 + 2.98e7iT - 7.60e17T^{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

−10.31154805259827144155676120010, −9.748049062959511276087869234277, −8.799224900658293753791874654514, −7.23408072821207522649738520209, −6.71486062792806669902208336948, −5.75484477363596413161859672056, −4.31700304951097708354172030324, −3.16338006505104848011558424396, −2.26665976963257994117132316743, −1.30629731636653329943209229737, 0.10471853590096997016870828664, 1.46391156797036519946732682541, 2.73010133464088960875989942505, 3.47681861682071425794492097278, 5.24976900165472995158969775315, 6.04662411234854492853661941134, 6.72702986638833352948209535572, 7.79010700834811811049428085279, 8.709655927996797144478215637882, 9.923125643522682322861349727697

Graph of the ZZ-function along the critical line