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Properties

Label 2-6525-1.1-c1-0-103
Degree 22
Conductor 65256525
Sign 11
Analytic cond. 52.102352.1023
Root an. cond. 7.218197.21819
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  + 2.43·2-s + 3.94·4-s − 3.59·7-s + 4.75·8-s + 6.58·11-s − 2.25·13-s − 8.77·14-s + 3.69·16-s − 0.256·17-s + 7.95·19-s + 16.0·22-s − 0.961·23-s − 5.50·26-s − 14.2·28-s + 29-s − 3.95·31-s − 0.498·32-s − 0.624·34-s + 3.76·37-s + 19.3·38-s + 5.00·41-s + 6.19·43-s + 26.0·44-s − 2.34·46-s + 7.02·47-s + 5.93·49-s − 8.90·52-s + ⋯
L(s)  = 1  + 1.72·2-s + 1.97·4-s − 1.35·7-s + 1.67·8-s + 1.98·11-s − 0.625·13-s − 2.34·14-s + 0.922·16-s − 0.0621·17-s + 1.82·19-s + 3.42·22-s − 0.200·23-s − 1.07·26-s − 2.68·28-s + 0.185·29-s − 0.709·31-s − 0.0881·32-s − 0.107·34-s + 0.619·37-s + 3.14·38-s + 0.781·41-s + 0.945·43-s + 3.92·44-s − 0.345·46-s + 1.02·47-s + 0.848·49-s − 1.23·52-s + ⋯

Functional equation

Λ(s)=(6525s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6525s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 65256525    =    32⋅52⋅293^{2} \cdot 5^{2} \cdot 29
Sign: 11
Analytic conductor: 52.102352.1023
Root analytic conductor: 7.218197.21819
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 6525, ( :1/2), 1)(2,\ 6525,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 5.8803922725.880392272
L(12)L(\frac12) ≈\approx 5.8803922725.880392272
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)
bad3 1 1
5 1 1
29 1−T 1 - T
good2 1−2.43T+2T2 1 - 2.43T + 2T^{2}
7 1+3.59T+7T2 1 + 3.59T + 7T^{2}
11 1−6.58T+11T2 1 - 6.58T + 11T^{2}
13 1+2.25T+13T2 1 + 2.25T + 13T^{2}
17 1+0.256T+17T2 1 + 0.256T + 17T^{2}
19 1−7.95T+19T2 1 - 7.95T + 19T^{2}
23 1+0.961T+23T2 1 + 0.961T + 23T^{2}
31 1+3.95T+31T2 1 + 3.95T + 31T^{2}
37 1−3.76T+37T2 1 - 3.76T + 37T^{2}
41 1−5.00T+41T2 1 - 5.00T + 41T^{2}
43 1−6.19T+43T2 1 - 6.19T + 43T^{2}
47 1−7.02T+47T2 1 - 7.02T + 47T^{2}
53 1+6.38T+53T2 1 + 6.38T + 53T^{2}
59 1+3.19T+59T2 1 + 3.19T + 59T^{2}
61 1−14.4T+61T2 1 - 14.4T + 61T^{2}
67 1−9.35T+67T2 1 - 9.35T + 67T^{2}
71 1+11.9T+71T2 1 + 11.9T + 71T^{2}
73 1−13.4T+73T2 1 - 13.4T + 73T^{2}
79 1+5.22T+79T2 1 + 5.22T + 79T^{2}
83 1+0.195T+83T2 1 + 0.195T + 83T^{2}
89 1+3.72T+89T2 1 + 3.72T + 89T^{2}
97 1−2.97T+97T2 1 - 2.97T + 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

−7.43328357034537422652993964035, −7.10127228212765051671473877902, −6.31715516608663255864007083989, −5.94134094733814321437273423122, −5.13740710906782927030263719131, −4.23389267909244924575983638435, −3.68011424542833352656275311143, −3.13243776850671181935393376869, −2.26184662368037310433049516375, −0.992921124185419194564028446224, 0.992921124185419194564028446224, 2.26184662368037310433049516375, 3.13243776850671181935393376869, 3.68011424542833352656275311143, 4.23389267909244924575983638435, 5.13740710906782927030263719131, 5.94134094733814321437273423122, 6.31715516608663255864007083989, 7.10127228212765051671473877902, 7.43328357034537422652993964035

Graph of the ZZ-function along the critical line