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Properties

Label 2-35e2-1.1-c3-0-11
Degree 22
Conductor 12251225
Sign 11
Analytic cond. 72.277372.2773
Root an. cond. 8.501608.50160
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 2·3-s + 4-s − 6·6-s + 21·8-s − 23·9-s − 45·11-s + 2·12-s − 59·13-s − 71·16-s + 54·17-s + 69·18-s − 121·19-s + 135·22-s − 69·23-s + 42·24-s + 177·26-s − 100·27-s − 162·29-s − 88·31-s + 45·32-s − 90·33-s − 162·34-s − 23·36-s + 259·37-s + 363·38-s − 118·39-s + ⋯
L(s)  = 1  − 1.06·2-s + 0.384·3-s + 1/8·4-s − 0.408·6-s + 0.928·8-s − 0.851·9-s − 1.23·11-s + 0.0481·12-s − 1.25·13-s − 1.10·16-s + 0.770·17-s + 0.903·18-s − 1.46·19-s + 1.30·22-s − 0.625·23-s + 0.357·24-s + 1.33·26-s − 0.712·27-s − 1.03·29-s − 0.509·31-s + 0.248·32-s − 0.474·33-s − 0.817·34-s − 0.106·36-s + 1.15·37-s + 1.54·38-s − 0.484·39-s + ⋯

Functional equation

Λ(s)=(1225s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1225s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12251225    =    52⋅725^{2} \cdot 7^{2}
Sign: 11
Analytic conductor: 72.277372.2773
Root analytic conductor: 8.501608.50160
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1225, ( :3/2), 1)(2,\ 1225,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.34475680410.3447568041
L(12)L(\frac12) ≈\approx 0.34475680410.3447568041
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 1
good2 1+3T+p3T2 1 + 3 T + p^{3} T^{2}
3 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
11 1+45T+p3T2 1 + 45 T + p^{3} T^{2}
13 1+59T+p3T2 1 + 59 T + p^{3} T^{2}
17 1−54T+p3T2 1 - 54 T + p^{3} T^{2}
19 1+121T+p3T2 1 + 121 T + p^{3} T^{2}
23 1+3pT+p3T2 1 + 3 p T + p^{3} T^{2}
29 1+162T+p3T2 1 + 162 T + p^{3} T^{2}
31 1+88T+p3T2 1 + 88 T + p^{3} T^{2}
37 1−7pT+p3T2 1 - 7 p T + p^{3} T^{2}
41 1−195T+p3T2 1 - 195 T + p^{3} T^{2}
43 1−286T+p3T2 1 - 286 T + p^{3} T^{2}
47 1+45T+p3T2 1 + 45 T + p^{3} T^{2}
53 1+597T+p3T2 1 + 597 T + p^{3} T^{2}
59 1+360T+p3T2 1 + 360 T + p^{3} T^{2}
61 1−392T+p3T2 1 - 392 T + p^{3} T^{2}
67 1−280T+p3T2 1 - 280 T + p^{3} T^{2}
71 1−48T+p3T2 1 - 48 T + p^{3} T^{2}
73 1+668T+p3T2 1 + 668 T + p^{3} T^{2}
79 1−782T+p3T2 1 - 782 T + p^{3} T^{2}
83 1+768T+p3T2 1 + 768 T + p^{3} T^{2}
89 1+1194T+p3T2 1 + 1194 T + p^{3} T^{2}
97 1+902T+p3T2 1 + 902 T + p^{3} 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

−9.376365677049136337030528622700, −8.507734516966498819698463759291, −7.82638894032474638578069066495, −7.43611706745052403283995209497, −6.03153630643110133771569337587, −5.13785186455935956075930780139, −4.15001330393488703196686324770, −2.76318250472831319611100182638, −1.99167026243351752391888723186, −0.32633284316797679934872944314, 0.32633284316797679934872944314, 1.99167026243351752391888723186, 2.76318250472831319611100182638, 4.15001330393488703196686324770, 5.13785186455935956075930780139, 6.03153630643110133771569337587, 7.43611706745052403283995209497, 7.82638894032474638578069066495, 8.507734516966498819698463759291, 9.376365677049136337030528622700

Graph of the ZZ-function along the critical line