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Properties

Label 2-2646-1.1-c1-0-51
Degree 22
Conductor 26462646
Sign −1-1
Analytic cond. 21.128421.1284
Root an. cond. 4.596564.59656
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 6·11-s + 5·13-s + 16-s − 6·17-s − 4·19-s − 6·22-s − 6·23-s − 5·25-s + 5·26-s − 6·29-s − 31-s + 32-s − 6·34-s − 37-s − 4·38-s + 6·41-s − 43-s − 6·44-s − 6·46-s + 6·47-s − 5·50-s + 5·52-s + 6·53-s − 6·58-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.80·11-s + 1.38·13-s + 1/4·16-s − 1.45·17-s − 0.917·19-s − 1.27·22-s − 1.25·23-s − 25-s + 0.980·26-s − 1.11·29-s − 0.179·31-s + 0.176·32-s − 1.02·34-s − 0.164·37-s − 0.648·38-s + 0.937·41-s − 0.152·43-s − 0.904·44-s − 0.884·46-s + 0.875·47-s − 0.707·50-s + 0.693·52-s + 0.824·53-s − 0.787·58-s + ⋯

Functional equation

Λ(s)=(2646s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2646s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 26462646    =    2⋅33⋅722 \cdot 3^{3} \cdot 7^{2}
Sign: −1-1
Analytic conductor: 21.128421.1284
Root analytic conductor: 4.596564.59656
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2646, ( :1/2), −1)(2,\ 2646,\ (\ :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)
bad2 1−T 1 - T
3 1 1
7 1 1
good5 1+pT2 1 + p T^{2}
11 1+6T+pT2 1 + 6 T + p T^{2}
13 1−5T+pT2 1 - 5 T + p T^{2}
17 1+6T+pT2 1 + 6 T + p T^{2}
19 1+4T+pT2 1 + 4 T + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1+6T+pT2 1 + 6 T + p T^{2}
31 1+T+pT2 1 + T + p T^{2}
37 1+T+pT2 1 + T + p T^{2}
41 1−6T+pT2 1 - 6 T + p T^{2}
43 1+T+pT2 1 + T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1+T+pT2 1 + T + p T^{2}
67 1+T+pT2 1 + T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1−2T+pT2 1 - 2 T + p T^{2}
79 1+T+pT2 1 + T + p T^{2}
83 1+6T+pT2 1 + 6 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1−17T+pT2 1 - 17 T + p 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

−8.322567467361658853622879025591, −7.75571829894035808861745121169, −6.83258442473794854766223031310, −5.91864842852306924737657425731, −5.53634730988965039358149929804, −4.34596400781076763888194778584, −3.84967222428535684217921793847, −2.60629849220782543841100988768, −1.91689542680922239943101317453, 0, 1.91689542680922239943101317453, 2.60629849220782543841100988768, 3.84967222428535684217921793847, 4.34596400781076763888194778584, 5.53634730988965039358149929804, 5.91864842852306924737657425731, 6.83258442473794854766223031310, 7.75571829894035808861745121169, 8.322567467361658853622879025591

Graph of the ZZ-function along the critical line