Properties

Label 2-1152-8.3-c0-0-0
Degree $2$
Conductor $1152$
Sign $-i$
Analytic cond. $0.574922$
Root an. cond. $0.758236$
Motivic weight $0$
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 − 3·25-s + 2i·29-s + 49-s − 2i·53-s + 2·73-s + 2·97-s − 2i·101-s + ⋯
L(s)  = 1  + 2i·5-s − 3·25-s + 2i·29-s + 49-s − 2i·53-s + 2·73-s + 2·97-s − 2i·101-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9859431593\)
\(L(\frac12)\) \(\approx\) \(0.9859431593\)
\(L(1)\) 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 \( 1 - 2iT - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 2iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 2T + T^{2} \)
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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

−10.32904315598308936279450850125, −9.596811998769285891963235777661, −8.507164368235856257029258774545, −7.48568691682640405444696572155, −6.91198719731158661318955679234, −6.23329559472587249838751815123, −5.19873261023162234974249814724, −3.77623447827385301035880408703, −3.08854663638726921451892984130, −2.05433328193680623125449754459, 0.918015980458153970151122476360, 2.21553977607805232904353463113, 3.90962520960734225613550064608, 4.60283690933234830142766355764, 5.45780422271770771077743201769, 6.21381971161003882077582006974, 7.63378908382220094757359811056, 8.187545809273683691433180738659, 9.059740003860128767748979958533, 9.515921858426045983602301046016

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