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Properties

Label 2-1152-8.5-c1-0-8
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  + 4·7-s + 4i·11-s + 4i·13-s + 2·17-s − 4i·19-s − 8·23-s + 5·25-s + 8i·29-s − 4·31-s + 4i·37-s + 6·41-s − 4i·43-s + 8·47-s + 9·49-s − 8i·53-s + ⋯
L(s)  = 1  + 1.51·7-s + 1.20i·11-s + 1.10i·13-s + 0.485·17-s − 0.917i·19-s − 1.66·23-s + 25-s + 1.48i·29-s − 0.718·31-s + 0.657i·37-s + 0.937·41-s − 0.609i·43-s + 1.16·47-s + 1.28·49-s − 1.09i·53-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

\(L(1)\) \(\approx\) \(1.881575433\)
\(L(\frac12)\) \(\approx\) \(1.881575433\)
\(L(\frac{3}{2})\) 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 \( 1 - 5T^{2} \)
7 \( 1 - 4T + 7T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 - 8iT - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 8iT - 53T^{2} \)
59 \( 1 + 12iT - 59T^{2} \)
61 \( 1 - 12iT - 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.890884584524078086460046111068, −9.016393465530618757925132762868, −8.295967730300905786661193294856, −7.35659034201045942356868869490, −6.81030811603774714024506636250, −5.43998307798370986961544031869, −4.72385712682161959854602407232, −3.98951474176203430726040510357, −2.33665932454297985572256525507, −1.48666305912036011624599909232, 0.911501807918419972467098233395, 2.24257550388304404591405537177, 3.52199049335282005802690539379, 4.48673801811247150812678591755, 5.64346789004486508333665991381, 5.97246099724709468743033533834, 7.63455940501931738165346753153, 7.966622569732101710547939492039, 8.650811057674037010870100150293, 9.757819236639717806212717933968

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