Properties

Label 2-1152-72.29-c0-0-1
Degree $2$
Conductor $1152$
Sign $0.342 + 0.939i$
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  + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + 1.73i·17-s − 1.73i·19-s + (0.5 − 0.866i)25-s − 0.999·27-s + (−0.499 − 0.866i)33-s + (−1.5 + 0.866i)41-s + (1.5 + 0.866i)43-s + (0.5 + 0.866i)49-s + (1.49 + 0.866i)51-s + (−1.49 − 0.866i)57-s + (−0.5 − 0.866i)59-s + (−1.5 + 0.866i)67-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + 1.73i·17-s − 1.73i·19-s + (0.5 − 0.866i)25-s − 0.999·27-s + (−0.499 − 0.866i)33-s + (−1.5 + 0.866i)41-s + (1.5 + 0.866i)43-s + (0.5 + 0.866i)49-s + (1.49 + 0.866i)51-s + (−1.49 − 0.866i)57-s + (−0.5 − 0.866i)59-s + (−1.5 + 0.866i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(0.574922\)
Root analytic conductor: \(0.758236\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :0),\ 0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.220803001\)
\(L(\frac12)\) \(\approx\) \(1.220803001\)
\(L(1)\) 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 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.73iT - T^{2} \)
19 \( 1 + 1.73iT - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.660693034100202583981091375850, −8.719966620729926208168790030478, −8.389269550096887448173989284572, −7.36376255195429644191716471930, −6.46729429166528441585294980128, −5.96271890973023822389230858360, −4.55014463992081048831084977891, −3.45136144260439441947776489926, −2.49722513095698334688082160395, −1.16313552105860819673581777060, 1.89784221832519258796381481516, 3.12486692057215219076220711185, 4.01774777797790538428691430007, 4.92844384708860131588241994054, 5.71830336706417071697583893626, 7.05928280792007312886934696096, 7.66951300868047803755641760938, 8.775020080361638186348592034790, 9.337354543190856321723673837020, 10.07443358411088344535432897769

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