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Properties

Label 2-7800-1.1-c1-0-10
Degree 22
Conductor 78007800
Sign 11
Analytic cond. 62.283362.2833
Root an. cond. 7.891977.89197
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 1.36·7-s + 9-s − 5.64·11-s + 13-s − 2.14·17-s + 2.41·19-s − 1.36·21-s + 2.72·23-s − 27-s − 5.28·29-s − 1.64·31-s + 5.64·33-s − 3.86·37-s − 39-s + 5.86·41-s + 9.00·43-s − 3.77·47-s − 5.14·49-s + 2.14·51-s − 12.0·53-s − 2.41·57-s + 6.19·59-s + 11.5·61-s + 1.36·63-s + 0.324·67-s − 2.72·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.515·7-s + 0.333·9-s − 1.70·11-s + 0.277·13-s − 0.519·17-s + 0.553·19-s − 0.297·21-s + 0.568·23-s − 0.192·27-s − 0.980·29-s − 0.295·31-s + 0.982·33-s − 0.635·37-s − 0.160·39-s + 0.916·41-s + 1.37·43-s − 0.551·47-s − 0.734·49-s + 0.299·51-s − 1.64·53-s − 0.319·57-s + 0.806·59-s + 1.48·61-s + 0.171·63-s + 0.0396·67-s − 0.328·69-s + ⋯

Functional equation

Λ(s)=(7800s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7800s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 78007800    =    23⋅3⋅52⋅132^{3} \cdot 3 \cdot 5^{2} \cdot 13
Sign: 11
Analytic conductor: 62.283362.2833
Root analytic conductor: 7.891977.89197
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7800, ( :1/2), 1)(2,\ 7800,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.2448976991.244897699
L(12)L(\frac12) ≈\approx 1.2448976991.244897699
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 1
3 1+T 1 + T
5 1 1
13 1−T 1 - T
good7 1−1.36T+7T2 1 - 1.36T + 7T^{2}
11 1+5.64T+11T2 1 + 5.64T + 11T^{2}
17 1+2.14T+17T2 1 + 2.14T + 17T^{2}
19 1−2.41T+19T2 1 - 2.41T + 19T^{2}
23 1−2.72T+23T2 1 - 2.72T + 23T^{2}
29 1+5.28T+29T2 1 + 5.28T + 29T^{2}
31 1+1.64T+31T2 1 + 1.64T + 31T^{2}
37 1+3.86T+37T2 1 + 3.86T + 37T^{2}
41 1−5.86T+41T2 1 - 5.86T + 41T^{2}
43 1−9.00T+43T2 1 - 9.00T + 43T^{2}
47 1+3.77T+47T2 1 + 3.77T + 47T^{2}
53 1+12.0T+53T2 1 + 12.0T + 53T^{2}
59 1−6.19T+59T2 1 - 6.19T + 59T^{2}
61 1−11.5T+61T2 1 - 11.5T + 61T^{2}
67 1−0.324T+67T2 1 - 0.324T + 67T^{2}
71 1−11.8T+71T2 1 - 11.8T + 71T^{2}
73 1+8.28T+73T2 1 + 8.28T + 73T^{2}
79 1−2.31T+79T2 1 - 2.31T + 79T^{2}
83 1+10.3T+83T2 1 + 10.3T + 83T^{2}
89 1−3.73T+89T2 1 - 3.73T + 89T^{2}
97 1−16.8T+97T2 1 - 16.8T + 97T^{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

−7.73392864804156021734863538023, −7.29944867645656172760258520513, −6.42371745310255394679206949906, −5.58505679949814194761449734491, −5.17324013302925341961322460105, −4.50360210309540243965134875292, −3.53551924642668006880083543848, −2.61678900211238879360640326458, −1.76774773119575064710442119044, −0.56446567111075060653312758195, 0.56446567111075060653312758195, 1.76774773119575064710442119044, 2.61678900211238879360640326458, 3.53551924642668006880083543848, 4.50360210309540243965134875292, 5.17324013302925341961322460105, 5.58505679949814194761449734491, 6.42371745310255394679206949906, 7.29944867645656172760258520513, 7.73392864804156021734863538023

Graph of the ZZ-function along the critical line