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Properties

Label 2-425-1.1-c1-0-3
Degree 22
Conductor 425425
Sign 11
Analytic cond. 3.393643.39364
Root an. cond. 1.842181.84218
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  − 2.60·2-s − 1.18·3-s + 4.77·4-s + 3.07·6-s + 3.53·7-s − 7.21·8-s − 1.60·9-s + 2.94·11-s − 5.64·12-s + 4.01·13-s − 9.20·14-s + 9.23·16-s − 17-s + 4.17·18-s − 6.97·19-s − 4.18·21-s − 7.66·22-s − 6.12·23-s + 8.53·24-s − 10.4·26-s + 5.44·27-s + 16.8·28-s + 5.30·29-s + 6.49·31-s − 9.59·32-s − 3.48·33-s + 2.60·34-s + ⋯
L(s)  = 1  − 1.84·2-s − 0.682·3-s + 2.38·4-s + 1.25·6-s + 1.33·7-s − 2.55·8-s − 0.534·9-s + 0.888·11-s − 1.62·12-s + 1.11·13-s − 2.45·14-s + 2.30·16-s − 0.242·17-s + 0.982·18-s − 1.60·19-s − 0.912·21-s − 1.63·22-s − 1.27·23-s + 1.74·24-s − 2.04·26-s + 1.04·27-s + 3.18·28-s + 0.984·29-s + 1.16·31-s − 1.69·32-s − 0.606·33-s + 0.446·34-s + ⋯

Functional equation

Λ(s)=(425s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(425s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 425425    =    52⋅175^{2} \cdot 17
Sign: 11
Analytic conductor: 3.393643.39364
Root analytic conductor: 1.842181.84218
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 425, ( :1/2), 1)(2,\ 425,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.54732072150.5473207215
L(12)L(\frac12) ≈\approx 0.54732072150.5473207215
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)
bad5 1 1
17 1+T 1 + T
good2 1+2.60T+2T2 1 + 2.60T + 2T^{2}
3 1+1.18T+3T2 1 + 1.18T + 3T^{2}
7 1−3.53T+7T2 1 - 3.53T + 7T^{2}
11 1−2.94T+11T2 1 - 2.94T + 11T^{2}
13 1−4.01T+13T2 1 - 4.01T + 13T^{2}
19 1+6.97T+19T2 1 + 6.97T + 19T^{2}
23 1+6.12T+23T2 1 + 6.12T + 23T^{2}
29 1−5.30T+29T2 1 - 5.30T + 29T^{2}
31 1−6.49T+31T2 1 - 6.49T + 31T^{2}
37 1−3.43T+37T2 1 - 3.43T + 37T^{2}
41 1−4.61T+41T2 1 - 4.61T + 41T^{2}
43 1−10.2T+43T2 1 - 10.2T + 43T^{2}
47 1−3.67T+47T2 1 - 3.67T + 47T^{2}
53 1+6.77T+53T2 1 + 6.77T + 53T^{2}
59 1−9.92T+59T2 1 - 9.92T + 59T^{2}
61 1+2.36T+61T2 1 + 2.36T + 61T^{2}
67 1+9.56T+67T2 1 + 9.56T + 67T^{2}
71 1−5.51T+71T2 1 - 5.51T + 71T^{2}
73 1−2.00T+73T2 1 - 2.00T + 73T^{2}
79 1−10.5T+79T2 1 - 10.5T + 79T^{2}
83 1−9.07T+83T2 1 - 9.07T + 83T^{2}
89 1−2.63T+89T2 1 - 2.63T + 89T^{2}
97 1−5.86T+97T2 1 - 5.86T + 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

−10.94805070342518662444957674120, −10.48113946114586763958242639779, −9.179829265380621468720940348191, −8.422462251088697546291467715765, −7.962402113983155982404201152036, −6.52136385484807824788455224299, −6.02983365902527318815063093902, −4.33516982401709700523998025002, −2.25703778911600787632433421835, −0.989009460232569672358177642769, 0.989009460232569672358177642769, 2.25703778911600787632433421835, 4.33516982401709700523998025002, 6.02983365902527318815063093902, 6.52136385484807824788455224299, 7.962402113983155982404201152036, 8.422462251088697546291467715765, 9.179829265380621468720940348191, 10.48113946114586763958242639779, 10.94805070342518662444957674120

Graph of the ZZ-function along the critical line