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Properties

Label 2-350-1.1-c5-0-24
Degree 22
Conductor 350350
Sign 11
Analytic cond. 56.134356.1343
Root an. cond. 7.492287.49228
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 27.5·3-s + 16·4-s − 110.·6-s + 49·7-s − 64·8-s + 514.·9-s + 566.·11-s + 440.·12-s − 238.·13-s − 196·14-s + 256·16-s + 854.·17-s − 2.05e3·18-s + 1.84e3·19-s + 1.34e3·21-s − 2.26e3·22-s − 4.12e3·23-s − 1.76e3·24-s + 955.·26-s + 7.47e3·27-s + 784·28-s − 7.22e3·29-s + 4.90e3·31-s − 1.02e3·32-s + 1.55e4·33-s − 3.41e3·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.76·3-s + 0.5·4-s − 1.24·6-s + 0.377·7-s − 0.353·8-s + 2.11·9-s + 1.41·11-s + 0.882·12-s − 0.391·13-s − 0.267·14-s + 0.250·16-s + 0.716·17-s − 1.49·18-s + 1.17·19-s + 0.667·21-s − 0.998·22-s − 1.62·23-s − 0.624·24-s + 0.277·26-s + 1.97·27-s + 0.188·28-s − 1.59·29-s + 0.916·31-s − 0.176·32-s + 2.49·33-s − 0.506·34-s + ⋯

Functional equation

Λ(s)=(350s/2 ΓC(s) L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(350s/2 ΓC(s+5/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 350350    =    2⋅52⋅72 \cdot 5^{2} \cdot 7
Sign: 11
Analytic conductor: 56.134356.1343
Root analytic conductor: 7.492287.49228
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 350, ( :5/2), 1)(2,\ 350,\ (\ :5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 3.5740947443.574094744
L(12)L(\frac12) ≈\approx 3.5740947443.574094744
L(72)L(\frac{7}{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+4T 1 + 4T
5 1 1
7 1−49T 1 - 49T
good3 1−27.5T+243T2 1 - 27.5T + 243T^{2}
11 1−566.T+1.61e5T2 1 - 566.T + 1.61e5T^{2}
13 1+238.T+3.71e5T2 1 + 238.T + 3.71e5T^{2}
17 1−854.T+1.41e6T2 1 - 854.T + 1.41e6T^{2}
19 1−1.84e3T+2.47e6T2 1 - 1.84e3T + 2.47e6T^{2}
23 1+4.12e3T+6.43e6T2 1 + 4.12e3T + 6.43e6T^{2}
29 1+7.22e3T+2.05e7T2 1 + 7.22e3T + 2.05e7T^{2}
31 1−4.90e3T+2.86e7T2 1 - 4.90e3T + 2.86e7T^{2}
37 1−3.40e3T+6.93e7T2 1 - 3.40e3T + 6.93e7T^{2}
41 1−1.78e4T+1.15e8T2 1 - 1.78e4T + 1.15e8T^{2}
43 1−3.41e3T+1.47e8T2 1 - 3.41e3T + 1.47e8T^{2}
47 1−1.32e4T+2.29e8T2 1 - 1.32e4T + 2.29e8T^{2}
53 1+2.31e4T+4.18e8T2 1 + 2.31e4T + 4.18e8T^{2}
59 1−1.81e4T+7.14e8T2 1 - 1.81e4T + 7.14e8T^{2}
61 1+3.28e4T+8.44e8T2 1 + 3.28e4T + 8.44e8T^{2}
67 1+6.60e4T+1.35e9T2 1 + 6.60e4T + 1.35e9T^{2}
71 1+1.09e4T+1.80e9T2 1 + 1.09e4T + 1.80e9T^{2}
73 1+3.94e4T+2.07e9T2 1 + 3.94e4T + 2.07e9T^{2}
79 1−6.14e4T+3.07e9T2 1 - 6.14e4T + 3.07e9T^{2}
83 1+5.60e3T+3.93e9T2 1 + 5.60e3T + 3.93e9T^{2}
89 1−1.26e5T+5.58e9T2 1 - 1.26e5T + 5.58e9T^{2}
97 1−1.38e5T+8.58e9T2 1 - 1.38e5T + 8.58e9T^{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

−10.20347647640354335138460066730, −9.410178445713308835027478743862, −8.990067507819972474415114932700, −7.73561182220425883398888906940, −7.55430450906983302458436309515, −6.06800771653844250880845935656, −4.24548235527092787267679474099, −3.31150454301165850009202475972, −2.11952965212645115672020278223, −1.16681719686071393847017741404, 1.16681719686071393847017741404, 2.11952965212645115672020278223, 3.31150454301165850009202475972, 4.24548235527092787267679474099, 6.06800771653844250880845935656, 7.55430450906983302458436309515, 7.73561182220425883398888906940, 8.990067507819972474415114932700, 9.410178445713308835027478743862, 10.20347647640354335138460066730

Graph of the ZZ-function along the critical line