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Properties

Label 2-315-35.34-c4-0-38
Degree 22
Conductor 315315
Sign 11
Analytic cond. 32.561532.5615
Root an. cond. 5.706275.70627
Motivic weight 44
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·4-s − 25·5-s + 49·7-s + 73·11-s + 23·13-s + 256·16-s − 263·17-s − 400·20-s + 625·25-s + 784·28-s + 1.15e3·29-s − 1.22e3·35-s + 1.16e3·44-s + 3.45e3·47-s + 2.40e3·49-s + 368·52-s − 1.82e3·55-s + 4.09e3·64-s − 575·65-s − 4.20e3·68-s + 1.00e4·71-s − 9.50e3·73-s + 3.57e3·77-s + 1.21e4·79-s − 6.40e3·80-s + 6.38e3·83-s + 6.57e3·85-s + ⋯
L(s)  = 1  + 4-s − 5-s + 7-s + 0.603·11-s + 0.136·13-s + 16-s − 0.910·17-s − 20-s + 25-s + 28-s + 1.37·29-s − 35-s + 0.603·44-s + 1.56·47-s + 49-s + 0.136·52-s − 0.603·55-s + 64-s − 0.136·65-s − 0.910·68-s + 1.99·71-s − 1.78·73-s + 0.603·77-s + 1.94·79-s − 80-s + 0.926·83-s + 0.910·85-s + ⋯

Functional equation

Λ(s)=(315s/2 ΓC(s) L(s)=( Λ(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
Λ(s)=(315s/2 ΓC(s+2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 315315    =    32⋅5⋅73^{2} \cdot 5 \cdot 7
Sign: 11
Analytic conductor: 32.561532.5615
Root analytic conductor: 5.706275.70627
Motivic weight: 44
Rational: yes
Arithmetic: yes
Character: χ315(244,⋅)\chi_{315} (244, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 315, ( :2), 1)(2,\ 315,\ (\ :2),\ 1)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 2.5292937292.529293729
L(12)L(\frac12) ≈\approx 2.5292937292.529293729
L(3)L(3) 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+p2T 1 + p^{2} T
7 1−p2T 1 - p^{2} T
good2 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
11 1−73T+p4T2 1 - 73 T + p^{4} T^{2}
13 1−23T+p4T2 1 - 23 T + p^{4} T^{2}
17 1+263T+p4T2 1 + 263 T + p^{4} T^{2}
19 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
23 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
29 1−1153T+p4T2 1 - 1153 T + p^{4} T^{2}
31 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
37 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
41 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
43 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
47 1−3457T+p4T2 1 - 3457 T + p^{4} T^{2}
53 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
59 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
61 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
67 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
71 1−10078T+p4T2 1 - 10078 T + p^{4} T^{2}
73 1+9502T+p4T2 1 + 9502 T + p^{4} T^{2}
79 1−12167T+p4T2 1 - 12167 T + p^{4} T^{2}
83 1−6382T+p4T2 1 - 6382 T + p^{4} T^{2}
89 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
97 1−3383T+p4T2 1 - 3383 T + p^{4} T^{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

−11.14169827501417819453778426814, −10.46862084264821887576153723480, −8.904788044812155669810465425491, −8.063583656677963155937712525887, −7.21254494712299882301260948405, −6.32109073254335625819948973452, −4.87494519743366996133717786272, −3.80023292654454020295098033991, −2.39948374559678777967862535598, −1.02430201843410146653340870684, 1.02430201843410146653340870684, 2.39948374559678777967862535598, 3.80023292654454020295098033991, 4.87494519743366996133717786272, 6.32109073254335625819948973452, 7.21254494712299882301260948405, 8.063583656677963155937712525887, 8.904788044812155669810465425491, 10.46862084264821887576153723480, 11.14169827501417819453778426814

Graph of the ZZ-function along the critical line