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Properties

Label 2-7650-1.1-c1-0-33
Degree 22
Conductor 76507650
Sign 11
Analytic cond. 61.085561.0855
Root an. cond. 7.815727.81572
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 7-s − 8-s + 3·11-s + 4·13-s + 14-s + 16-s + 17-s − 5·19-s − 3·22-s + 8·23-s − 4·26-s − 28-s + 4·29-s − 3·31-s − 32-s − 34-s + 7·37-s + 5·38-s − 2·41-s + 43-s + 3·44-s − 8·46-s − 7·47-s − 6·49-s + 4·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.904·11-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.242·17-s − 1.14·19-s − 0.639·22-s + 1.66·23-s − 0.784·26-s − 0.188·28-s + 0.742·29-s − 0.538·31-s − 0.176·32-s − 0.171·34-s + 1.15·37-s + 0.811·38-s − 0.312·41-s + 0.152·43-s + 0.452·44-s − 1.17·46-s − 1.02·47-s − 6/7·49-s + 0.554·52-s + ⋯

Functional equation

Λ(s)=(7650s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7650s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 76507650    =    2⋅32⋅52⋅172 \cdot 3^{2} \cdot 5^{2} \cdot 17
Sign: 11
Analytic conductor: 61.085561.0855
Root analytic conductor: 7.815727.81572
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7650, ( :1/2), 1)(2,\ 7650,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.5911465931.591146593
L(12)L(\frac12) ≈\approx 1.5911465931.591146593
L(32)L(\frac{3}{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+T 1 + T
3 1 1
5 1 1
17 1−T 1 - T
good7 1+T+pT2 1 + T + p T^{2}
11 1−3T+pT2 1 - 3 T + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
19 1+5T+pT2 1 + 5 T + p T^{2}
23 1−8T+pT2 1 - 8 T + p T^{2}
29 1−4T+pT2 1 - 4 T + p T^{2}
31 1+3T+pT2 1 + 3 T + p T^{2}
37 1−7T+pT2 1 - 7 T + p T^{2}
41 1+2T+pT2 1 + 2 T + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1+7T+pT2 1 + 7 T + p T^{2}
53 1−7T+pT2 1 - 7 T + p T^{2}
59 1−8T+pT2 1 - 8 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−11T+pT2 1 - 11 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+15T+pT2 1 + 15 T + p T^{2}
83 1−16T+pT2 1 - 16 T + p T^{2}
89 1+2T+pT2 1 + 2 T + p T^{2}
97 1+10T+pT2 1 + 10 T + p 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

−8.135292844046089664559906175869, −7.01740364334808750521314711647, −6.63898337253409794146618749411, −6.05399666296688153162808443071, −5.14787043994372317388094418023, −4.15843427047876882349228250562, −3.47462301731735252674209775046, −2.62891927239608265743979570517, −1.54999844370951157077561065508, −0.74929188852310592680180496602, 0.74929188852310592680180496602, 1.54999844370951157077561065508, 2.62891927239608265743979570517, 3.47462301731735252674209775046, 4.15843427047876882349228250562, 5.14787043994372317388094418023, 6.05399666296688153162808443071, 6.63898337253409794146618749411, 7.01740364334808750521314711647, 8.135292844046089664559906175869

Graph of the ZZ-function along the critical line