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Properties

Label 2-6525-1.1-c1-0-118
Degree 22
Conductor 65256525
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.66·2-s + 5.09·4-s − 0.0170·7-s − 8.24·8-s − 1.70·11-s − 4.69·13-s + 0.0453·14-s + 11.7·16-s + 3.91·17-s + 6.02·19-s + 4.55·22-s − 4.82·23-s + 12.5·26-s − 0.0868·28-s + 29-s + 1.33·31-s − 14.8·32-s − 10.4·34-s − 8.18·37-s − 16.0·38-s − 8.36·41-s + 3.72·43-s − 8.71·44-s + 12.8·46-s + 5.36·47-s − 6.99·49-s − 23.9·52-s + ⋯
L(s)  = 1  − 1.88·2-s + 2.54·4-s − 0.00644·7-s − 2.91·8-s − 0.515·11-s − 1.30·13-s + 0.0121·14-s + 2.94·16-s + 0.949·17-s + 1.38·19-s + 0.971·22-s − 1.00·23-s + 2.45·26-s − 0.0164·28-s + 0.185·29-s + 0.240·31-s − 2.62·32-s − 1.78·34-s − 1.34·37-s − 2.60·38-s − 1.30·41-s + 0.568·43-s − 1.31·44-s + 1.89·46-s + 0.783·47-s − 0.999·49-s − 3.31·52-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: −1-1
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: 11
Selberg data: (2, 6525, ( :1/2), −1)(2,\ 6525,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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+2.66T+2T2 1 + 2.66T + 2T^{2}
7 1+0.0170T+7T2 1 + 0.0170T + 7T^{2}
11 1+1.70T+11T2 1 + 1.70T + 11T^{2}
13 1+4.69T+13T2 1 + 4.69T + 13T^{2}
17 1−3.91T+17T2 1 - 3.91T + 17T^{2}
19 1−6.02T+19T2 1 - 6.02T + 19T^{2}
23 1+4.82T+23T2 1 + 4.82T + 23T^{2}
31 1−1.33T+31T2 1 - 1.33T + 31T^{2}
37 1+8.18T+37T2 1 + 8.18T + 37T^{2}
41 1+8.36T+41T2 1 + 8.36T + 41T^{2}
43 1−3.72T+43T2 1 - 3.72T + 43T^{2}
47 1−5.36T+47T2 1 - 5.36T + 47T^{2}
53 1−7.21T+53T2 1 - 7.21T + 53T^{2}
59 1−8.65T+59T2 1 - 8.65T + 59T^{2}
61 1−5.72T+61T2 1 - 5.72T + 61T^{2}
67 1−3.87T+67T2 1 - 3.87T + 67T^{2}
71 1−4.32T+71T2 1 - 4.32T + 71T^{2}
73 1+1.78T+73T2 1 + 1.78T + 73T^{2}
79 1−0.233T+79T2 1 - 0.233T + 79T^{2}
83 1−6.15T+83T2 1 - 6.15T + 83T^{2}
89 1−12.4T+89T2 1 - 12.4T + 89T^{2}
97 1+19.2T+97T2 1 + 19.2T + 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

−7.915806525853317785872822875458, −7.16730154904793843924436211815, −6.73208390106931941567976193168, −5.61722730039335985952213258582, −5.13645935098172648585929567846, −3.65092038505664532493467402809, −2.77742429154066799543385515355, −2.06617932622121829615956977042, −1.05002558935348783221617270892, 0, 1.05002558935348783221617270892, 2.06617932622121829615956977042, 2.77742429154066799543385515355, 3.65092038505664532493467402809, 5.13645935098172648585929567846, 5.61722730039335985952213258582, 6.73208390106931941567976193168, 7.16730154904793843924436211815, 7.915806525853317785872822875458

Graph of the ZZ-function along the critical line