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Properties

Label 2-175-1.1-c3-0-5
Degree 22
Conductor 175175
Sign 11
Analytic cond. 10.325310.3253
Root an. cond. 3.213303.21330
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.504·2-s − 4.26·3-s − 7.74·4-s − 2.15·6-s − 7·7-s − 7.94·8-s − 8.82·9-s + 54.8·11-s + 33.0·12-s − 16.0·13-s − 3.53·14-s + 57.9·16-s + 0.422·17-s − 4.45·18-s + 127.·19-s + 29.8·21-s + 27.7·22-s − 51.1·23-s + 33.8·24-s − 8.08·26-s + 152.·27-s + 54.2·28-s + 41.4·29-s + 192.·31-s + 92.8·32-s − 233.·33-s + 0.213·34-s + ⋯
L(s)  = 1  + 0.178·2-s − 0.820·3-s − 0.968·4-s − 0.146·6-s − 0.377·7-s − 0.351·8-s − 0.326·9-s + 1.50·11-s + 0.794·12-s − 0.341·13-s − 0.0674·14-s + 0.905·16-s + 0.00602·17-s − 0.0583·18-s + 1.53·19-s + 0.310·21-s + 0.268·22-s − 0.463·23-s + 0.288·24-s − 0.0609·26-s + 1.08·27-s + 0.365·28-s + 0.265·29-s + 1.11·31-s + 0.512·32-s − 1.23·33-s + 0.00107·34-s + ⋯

Functional equation

Λ(s)=(175s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(175s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 175175    =    52⋅75^{2} \cdot 7
Sign: 11
Analytic conductor: 10.325310.3253
Root analytic conductor: 3.213303.21330
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 175, ( :3/2), 1)(2,\ 175,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.97890070940.9789007094
L(12)L(\frac12) ≈\approx 0.97890070940.9789007094
L(52)L(\frac{5}{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)
bad5 1 1
7 1+7T 1 + 7T
good2 1−0.504T+8T2 1 - 0.504T + 8T^{2}
3 1+4.26T+27T2 1 + 4.26T + 27T^{2}
11 1−54.8T+1.33e3T2 1 - 54.8T + 1.33e3T^{2}
13 1+16.0T+2.19e3T2 1 + 16.0T + 2.19e3T^{2}
17 1−0.422T+4.91e3T2 1 - 0.422T + 4.91e3T^{2}
19 1−127.T+6.85e3T2 1 - 127.T + 6.85e3T^{2}
23 1+51.1T+1.21e4T2 1 + 51.1T + 1.21e4T^{2}
29 1−41.4T+2.43e4T2 1 - 41.4T + 2.43e4T^{2}
31 1−192.T+2.97e4T2 1 - 192.T + 2.97e4T^{2}
37 1−189.T+5.06e4T2 1 - 189.T + 5.06e4T^{2}
41 1+76.3T+6.89e4T2 1 + 76.3T + 6.89e4T^{2}
43 1+294.T+7.95e4T2 1 + 294.T + 7.95e4T^{2}
47 1+540.T+1.03e5T2 1 + 540.T + 1.03e5T^{2}
53 1−661.T+1.48e5T2 1 - 661.T + 1.48e5T^{2}
59 1−410.T+2.05e5T2 1 - 410.T + 2.05e5T^{2}
61 1−46.0T+2.26e5T2 1 - 46.0T + 2.26e5T^{2}
67 1+10.4T+3.00e5T2 1 + 10.4T + 3.00e5T^{2}
71 1+491.T+3.57e5T2 1 + 491.T + 3.57e5T^{2}
73 1−814.T+3.89e5T2 1 - 814.T + 3.89e5T^{2}
79 1+858.T+4.93e5T2 1 + 858.T + 4.93e5T^{2}
83 1−1.05e3T+5.71e5T2 1 - 1.05e3T + 5.71e5T^{2}
89 1−341.T+7.04e5T2 1 - 341.T + 7.04e5T^{2}
97 1−1.41e3T+9.12e5T2 1 - 1.41e3T + 9.12e5T^{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

−11.94407056366102801462090596588, −11.72068875396477307731707721648, −10.10584177281600404172853271759, −9.370374024224624385363332349352, −8.289592867360317828739872464393, −6.73154344767138228439247298320, −5.73767964529781460104708135065, −4.67579796851340841731909011122, −3.38116872597316928849224752835, −0.810811962771265949365583456405, 0.810811962771265949365583456405, 3.38116872597316928849224752835, 4.67579796851340841731909011122, 5.73767964529781460104708135065, 6.73154344767138228439247298320, 8.289592867360317828739872464393, 9.370374024224624385363332349352, 10.10584177281600404172853271759, 11.72068875396477307731707721648, 11.94407056366102801462090596588

Graph of the ZZ-function along the critical line