Properties

Label 2-1152-8.5-c1-0-11
Degree $2$
Conductor $1152$
Sign $0.707 + 0.707i$
Analytic cond. $9.19876$
Root an. cond. $3.03294$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·5-s + 4i·13-s + 8·17-s + 25-s − 10i·29-s − 12i·37-s + 8·41-s − 7·49-s − 14i·53-s + 12i·61-s + 8·65-s − 6·73-s − 16i·85-s + 16·89-s + 18·97-s + ⋯
L(s)  = 1  − 0.894i·5-s + 1.10i·13-s + 1.94·17-s + 0.200·25-s − 1.85i·29-s − 1.97i·37-s + 1.24·41-s − 49-s − 1.92i·53-s + 1.53i·61-s + 0.992·65-s − 0.702·73-s − 1.73i·85-s + 1.69·89-s + 1.82·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.675345593\)
\(L(\frac12)\) \(\approx\) \(1.675345593\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + 2iT - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 8T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 10iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 12iT - 37T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 14iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 12iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 16T + 89T^{2} \)
97 \( 1 - 18T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.569195906209353316386103391816, −8.992160693593646143980419697927, −8.032695675415850048715968186992, −7.38636052496249254937044044400, −6.20441728924070196660451788414, −5.43560605668167663455910920234, −4.48912698965887130754000845781, −3.62316112592333676467084540495, −2.17637646862648988184829450490, −0.883939896211164024581406165787, 1.23654910196504229459775032339, 2.96975377663197412870246570883, 3.33587521588229330439508665186, 4.85520788150980356817474621153, 5.70426406218297600320158904448, 6.57431551751645437142274452492, 7.52058281928250969497260568069, 8.054031608146775213709927224735, 9.154023296156999509157275982202, 10.13852419384693859791938650779

Graph of the $Z$-function along the critical line