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Properties

Label 2-1152-9.7-c1-0-5
Degree $2$
Conductor $1152$
Sign $-0.640 - 0.767i$
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  + (−1.57 + 0.729i)3-s + (−0.115 − 0.200i)5-s + (−0.230 + 0.399i)7-s + (1.93 − 2.29i)9-s + (−0.749 + 1.29i)11-s + (1.07 + 1.85i)13-s + (0.328 + 0.230i)15-s + 1.03·17-s − 2.94·19-s + (0.0707 − 0.796i)21-s + (−0.364 − 0.631i)23-s + (2.47 − 4.28i)25-s + (−1.36 + 5.01i)27-s + (−2.33 + 4.04i)29-s + (2.73 + 4.73i)31-s + ⋯
L(s)  = 1  + (−0.906 + 0.421i)3-s + (−0.0518 − 0.0897i)5-s + (−0.0872 + 0.151i)7-s + (0.644 − 0.764i)9-s + (−0.225 + 0.391i)11-s + (0.296 + 0.514i)13-s + (0.0847 + 0.0595i)15-s + 0.251·17-s − 0.675·19-s + (0.0154 − 0.173i)21-s + (−0.0760 − 0.131i)23-s + (0.494 − 0.856i)25-s + (−0.262 + 0.964i)27-s + (−0.433 + 0.751i)29-s + (0.491 + 0.851i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 - 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.640 - 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.640 - 0.767i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.640 - 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6842585402\)
\(L(\frac12)\) \(\approx\) \(0.6842585402\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.57 - 0.729i)T \)
good5 \( 1 + (0.115 + 0.200i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.230 - 0.399i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.749 - 1.29i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.07 - 1.85i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.03T + 17T^{2} \)
19 \( 1 + 2.94T + 19T^{2} \)
23 \( 1 + (0.364 + 0.631i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.33 - 4.04i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.73 - 4.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.30T + 37T^{2} \)
41 \( 1 + (1.84 + 3.18i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.41 - 4.19i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.40 - 9.36i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 10.0T + 53T^{2} \)
59 \( 1 + (-2.71 - 4.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.86 - 11.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.58 + 9.67i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.2T + 71T^{2} \)
73 \( 1 - 4.24T + 73T^{2} \)
79 \( 1 + (6.23 - 10.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.62 + 6.28i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + (2.21 - 3.83i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.23922514976248923359990743756, −9.354703111127234923404353494242, −8.615311400358598117957141058862, −7.49850920721549049166569387564, −6.56715975585031355817695897082, −5.94974463794698805200763572629, −4.84142542762576284922223704731, −4.27703242098866792244104013323, −2.99507724242045059902487234785, −1.40851980632449211888189076048, 0.35103356654322131478356691539, 1.79583791238134322887931577220, 3.21861364727229155540008341380, 4.39711247329501447560420411438, 5.40050128542928910997183640105, 6.09531660701493654719542789679, 6.92399876825117326227090331644, 7.78898512187415967575285702708, 8.500612888689032358682751568094, 9.738482636142613224962697212829

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