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Properties

Label 2-2535-1.1-c1-0-88
Degree 22
Conductor 25352535
Sign −1-1
Analytic cond. 20.242020.2420
Root an. cond. 4.499114.49911
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  − 1.80·2-s + 3-s + 1.24·4-s − 5-s − 1.80·6-s + 2.80·7-s + 1.35·8-s + 9-s + 1.80·10-s + 3.49·11-s + 1.24·12-s − 5.04·14-s − 15-s − 4.93·16-s − 7.60·17-s − 1.80·18-s + 1.75·19-s − 1.24·20-s + 2.80·21-s − 6.29·22-s − 6.44·23-s + 1.35·24-s + 25-s + 27-s + 3.49·28-s − 9.74·29-s + 1.80·30-s + ⋯
L(s)  = 1  − 1.27·2-s + 0.577·3-s + 0.623·4-s − 0.447·5-s − 0.735·6-s + 1.05·7-s + 0.479·8-s + 0.333·9-s + 0.569·10-s + 1.05·11-s + 0.359·12-s − 1.34·14-s − 0.258·15-s − 1.23·16-s − 1.84·17-s − 0.424·18-s + 0.402·19-s − 0.278·20-s + 0.611·21-s − 1.34·22-s − 1.34·23-s + 0.276·24-s + 0.200·25-s + 0.192·27-s + 0.660·28-s − 1.80·29-s + 0.328·30-s + ⋯

Functional equation

Λ(s)=(2535s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2535s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25352535    =    3⋅5⋅1323 \cdot 5 \cdot 13^{2}
Sign: −1-1
Analytic conductor: 20.242020.2420
Root analytic conductor: 4.499114.49911
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2535, ( :1/2), −1)(2,\ 2535,\ (\ :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−T 1 - T
5 1+T 1 + T
13 1 1
good2 1+1.80T+2T2 1 + 1.80T + 2T^{2}
7 1−2.80T+7T2 1 - 2.80T + 7T^{2}
11 1−3.49T+11T2 1 - 3.49T + 11T^{2}
17 1+7.60T+17T2 1 + 7.60T + 17T^{2}
19 1−1.75T+19T2 1 - 1.75T + 19T^{2}
23 1+6.44T+23T2 1 + 6.44T + 23T^{2}
29 1+9.74T+29T2 1 + 9.74T + 29T^{2}
31 1+9.59T+31T2 1 + 9.59T + 31T^{2}
37 1+6.85T+37T2 1 + 6.85T + 37T^{2}
41 1+1.19T+41T2 1 + 1.19T + 41T^{2}
43 1−4.00T+43T2 1 - 4.00T + 43T^{2}
47 1+2.97T+47T2 1 + 2.97T + 47T^{2}
53 1−4.51T+53T2 1 - 4.51T + 53T^{2}
59 1+7.18T+59T2 1 + 7.18T + 59T^{2}
61 1−4.43T+61T2 1 - 4.43T + 61T^{2}
67 1+2.45T+67T2 1 + 2.45T + 67T^{2}
71 1−13.5T+71T2 1 - 13.5T + 71T^{2}
73 1−1.75T+73T2 1 - 1.75T + 73T^{2}
79 1−4.85T+79T2 1 - 4.85T + 79T^{2}
83 1+12.0T+83T2 1 + 12.0T + 83T^{2}
89 1+4.12T+89T2 1 + 4.12T + 89T^{2}
97 1−2.32T+97T2 1 - 2.32T + 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

−8.619073142407194497215766797107, −7.953581274842170924514234964575, −7.31434219702273036526518342219, −6.67082930258912340249350863303, −5.34662675583794823025150126557, −4.28413405190741872460227047411, −3.78978090977089315611512289330, −2.08541921595419328049380585429, −1.59779431297846226125231121597, 0, 1.59779431297846226125231121597, 2.08541921595419328049380585429, 3.78978090977089315611512289330, 4.28413405190741872460227047411, 5.34662675583794823025150126557, 6.67082930258912340249350863303, 7.31434219702273036526518342219, 7.953581274842170924514234964575, 8.619073142407194497215766797107

Graph of the ZZ-function along the critical line