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Properties

Label 2-175-1.1-c3-0-12
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  + 4.84·2-s − 9.58·3-s + 15.4·4-s − 46.3·6-s + 7·7-s + 36.0·8-s + 64.7·9-s + 62.1·11-s − 147.·12-s + 14.0·13-s + 33.8·14-s + 50.9·16-s + 63.5·17-s + 313.·18-s + 48.7·19-s − 67.0·21-s + 301.·22-s − 99.3·23-s − 345.·24-s + 68.2·26-s − 362.·27-s + 108.·28-s − 69.0·29-s − 9.68·31-s − 41.7·32-s − 595.·33-s + 307.·34-s + ⋯
L(s)  = 1  + 1.71·2-s − 1.84·3-s + 1.93·4-s − 3.15·6-s + 0.377·7-s + 1.59·8-s + 2.39·9-s + 1.70·11-s − 3.55·12-s + 0.300·13-s + 0.646·14-s + 0.795·16-s + 0.906·17-s + 4.10·18-s + 0.588·19-s − 0.696·21-s + 2.91·22-s − 0.900·23-s − 2.93·24-s + 0.514·26-s − 2.58·27-s + 0.729·28-s − 0.442·29-s − 0.0561·31-s − 0.230·32-s − 3.14·33-s + 1.55·34-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 2.9657416822.965741682
L(12)L(\frac12) ≈\approx 2.9657416822.965741682
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 1−4.84T+8T2 1 - 4.84T + 8T^{2}
3 1+9.58T+27T2 1 + 9.58T + 27T^{2}
11 1−62.1T+1.33e3T2 1 - 62.1T + 1.33e3T^{2}
13 1−14.0T+2.19e3T2 1 - 14.0T + 2.19e3T^{2}
17 1−63.5T+4.91e3T2 1 - 63.5T + 4.91e3T^{2}
19 1−48.7T+6.85e3T2 1 - 48.7T + 6.85e3T^{2}
23 1+99.3T+1.21e4T2 1 + 99.3T + 1.21e4T^{2}
29 1+69.0T+2.43e4T2 1 + 69.0T + 2.43e4T^{2}
31 1+9.68T+2.97e4T2 1 + 9.68T + 2.97e4T^{2}
37 1−240.T+5.06e4T2 1 - 240.T + 5.06e4T^{2}
41 1−335.T+6.89e4T2 1 - 335.T + 6.89e4T^{2}
43 1−51.2T+7.95e4T2 1 - 51.2T + 7.95e4T^{2}
47 1+451.T+1.03e5T2 1 + 451.T + 1.03e5T^{2}
53 1+180.T+1.48e5T2 1 + 180.T + 1.48e5T^{2}
59 1−268.T+2.05e5T2 1 - 268.T + 2.05e5T^{2}
61 1+323.T+2.26e5T2 1 + 323.T + 2.26e5T^{2}
67 1+541.T+3.00e5T2 1 + 541.T + 3.00e5T^{2}
71 1+161.T+3.57e5T2 1 + 161.T + 3.57e5T^{2}
73 1−305.T+3.89e5T2 1 - 305.T + 3.89e5T^{2}
79 1+504.T+4.93e5T2 1 + 504.T + 4.93e5T^{2}
83 1+513.T+5.71e5T2 1 + 513.T + 5.71e5T^{2}
89 1−543.T+7.04e5T2 1 - 543.T + 7.04e5T^{2}
97 1−1.86e3T+9.12e5T2 1 - 1.86e3T + 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

−12.10037201381193873882679650320, −11.65505903800325254953813146299, −10.98230026189314591039648801696, −9.665859362322318616591747280309, −7.40112779910478852028997592301, −6.30302105881551788717011409308, −5.79567778775109122752182892771, −4.68188531151766533414820375917, −3.80684917722860389696762232536, −1.35796682910238191542612485570, 1.35796682910238191542612485570, 3.80684917722860389696762232536, 4.68188531151766533414820375917, 5.79567778775109122752182892771, 6.30302105881551788717011409308, 7.40112779910478852028997592301, 9.665859362322318616591747280309, 10.98230026189314591039648801696, 11.65505903800325254953813146299, 12.10037201381193873882679650320

Graph of the ZZ-function along the critical line