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Properties

Label 2-1152-72.59-c1-0-0
Degree $2$
Conductor $1152$
Sign $-0.703 + 0.710i$
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  + (0.524 + 1.65i)3-s + (−0.158 + 0.275i)5-s + (−2.93 + 1.69i)7-s + (−2.44 + 1.73i)9-s + (−0.642 + 0.370i)11-s + (−2.59 − 1.5i)13-s + (−0.537 − 0.117i)15-s − 4.24i·17-s + 2.57·19-s + (−4.33 − 3.94i)21-s + (−2.02 + 3.50i)23-s + (2.44 + 4.24i)25-s + (−4.14 − 3.13i)27-s + (−4.01 − 6.94i)29-s + (−4.24 − 2.45i)31-s + ⋯
L(s)  = 1  + (0.302 + 0.953i)3-s + (−0.0710 + 0.123i)5-s + (−1.10 + 0.639i)7-s + (−0.816 + 0.577i)9-s + (−0.193 + 0.111i)11-s + (−0.720 − 0.416i)13-s + (−0.138 − 0.0304i)15-s − 1.02i·17-s + 0.589·19-s + (−0.944 − 0.861i)21-s + (−0.421 + 0.730i)23-s + (0.489 + 0.848i)25-s + (−0.797 − 0.603i)27-s + (−0.745 − 1.29i)29-s + (−0.762 − 0.440i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.703 + 0.710i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.703 + 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2198875024\)
\(L(\frac12)\) \(\approx\) \(0.2198875024\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.524 - 1.65i)T \)
good5 \( 1 + (0.158 - 0.275i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.93 - 1.69i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.642 - 0.370i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.59 + 1.5i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.24iT - 17T^{2} \)
19 \( 1 - 2.57T + 19T^{2} \)
23 \( 1 + (2.02 - 3.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.01 + 6.94i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.24 + 2.45i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 7.34iT - 37T^{2} \)
41 \( 1 + (0.825 + 0.476i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.50 + 6.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.15 - 8.93i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 5.51T + 53T^{2} \)
59 \( 1 + (-3.50 - 2.02i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.79 - 4.5i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.36 - 11.0i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 + 6.44T + 73T^{2} \)
79 \( 1 + (8.29 - 4.78i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.21 + 1.27i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 14.6iT - 89T^{2} \)
97 \( 1 + (5.62 + 9.74i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.03211249795955319644306429990, −9.452239123455915242659273814435, −9.094353807986869303957223496509, −7.80843126575936301507713893420, −7.16455106459912379044976874080, −5.76880296513533134398209166155, −5.35752790848117111546703350766, −4.11220682752026414414706939678, −3.16003319642072611547237912123, −2.44226915846187607035792148125, 0.086381281658737806642874781145, 1.61204094178272971763861801868, 2.91534296714422908297021598484, 3.72966122863632018348368493468, 5.02856043648092414645058072476, 6.25710468776962212859401249951, 6.76049584197008466592636180604, 7.55471460297726334478459002818, 8.400777942779318507598793112326, 9.206815087126266758741776416012

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