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Properties

Label 2-5376-8.5-c1-0-67
Degree 22
Conductor 53765376
Sign −0.707+0.707i-0.707 + 0.707i
Analytic cond. 42.927542.9275
Root an. cond. 6.551916.55191
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s − 2i·5-s − 7-s − 9-s + 4i·11-s + 6i·13-s + 2·15-s − 2·17-s + 4i·19-s − i·21-s − 4·23-s + 25-s − i·27-s + 2i·29-s − 8·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.894i·5-s − 0.377·7-s − 0.333·9-s + 1.20i·11-s + 1.66i·13-s + 0.516·15-s − 0.485·17-s + 0.917i·19-s − 0.218i·21-s − 0.834·23-s + 0.200·25-s − 0.192i·27-s + 0.371i·29-s − 1.43·31-s + ⋯

Functional equation

Λ(s)=(5376s/2 ΓC(s) L(s)=((−0.707+0.707i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(5376s/2 ΓC(s+1/2) L(s)=((−0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 53765376    =    28⋅3⋅72^{8} \cdot 3 \cdot 7
Sign: −0.707+0.707i-0.707 + 0.707i
Analytic conductor: 42.927542.9275
Root analytic conductor: 6.551916.55191
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ5376(2689,⋅)\chi_{5376} (2689, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 5376, ( :1/2), −0.707+0.707i)(2,\ 5376,\ (\ :1/2),\ -0.707 + 0.707i)

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 1
3 1−iT 1 - iT
7 1+T 1 + T
good5 1+2iT−5T2 1 + 2iT - 5T^{2}
11 1−4iT−11T2 1 - 4iT - 11T^{2}
13 1−6iT−13T2 1 - 6iT - 13T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 1−4iT−19T2 1 - 4iT - 19T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1−2iT−29T2 1 - 2iT - 29T^{2}
31 1+8T+31T2 1 + 8T + 31T^{2}
37 1+10iT−37T2 1 + 10iT - 37T^{2}
41 1−2T+41T2 1 - 2T + 41T^{2}
43 1+8iT−43T2 1 + 8iT - 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1+10iT−53T2 1 + 10iT - 53T^{2}
59 1−12iT−59T2 1 - 12iT - 59T^{2}
61 1+10iT−61T2 1 + 10iT - 61T^{2}
67 1+8iT−67T2 1 + 8iT - 67T^{2}
71 1−12T+71T2 1 - 12T + 71T^{2}
73 1+2T+73T2 1 + 2T + 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1−12iT−83T2 1 - 12iT - 83T^{2}
89 1+6T+89T2 1 + 6T + 89T^{2}
97 1−2T+97T2 1 - 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

−7.980429295829972763339150929376, −7.12848448210342444478720069643, −6.55197057222954709846150037150, −5.54797915635949854005537895631, −4.96385823607857016413063572385, −4.01473040705127168586474644964, −3.88383371842442626249107445350, −2.25480581246777942234856344863, −1.67371036811037920712114210785, 0, 1.07402376152324713741471343362, 2.54705289132506810292487891514, 2.98805194554284973854058449427, 3.71516227340279091913054824687, 4.95201737198047055716805399331, 5.83618228497125908807436319511, 6.26426247803484509552425333969, 7.00005604742938908064987684111, 7.72213915376762324295688574673

Graph of the ZZ-function along the critical line