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Properties

Label 2-1152-72.59-c1-0-30
Degree $2$
Conductor $1152$
Sign $0.710 + 0.703i$
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  + (−1.65 + 0.524i)3-s + (1.01 − 1.75i)5-s + (4.09 − 2.36i)7-s + (2.44 − 1.73i)9-s + (−2.93 + 1.69i)11-s + (5.51 + 3.18i)13-s + (−0.751 + 3.42i)15-s − 4.87i·17-s + 1.48·19-s + (−5.51 + 6.04i)21-s + (−0.751 + 1.30i)23-s + (0.449 + 0.778i)25-s + (−3.13 + 4.14i)27-s + (−3.49 − 6.04i)29-s + (−5.93 − 3.42i)31-s + ⋯
L(s)  = 1  + (−0.953 + 0.302i)3-s + (0.452 − 0.784i)5-s + (1.54 − 0.893i)7-s + (0.816 − 0.577i)9-s + (−0.883 + 0.510i)11-s + (1.53 + 0.883i)13-s + (−0.193 + 0.884i)15-s − 1.18i·17-s + 0.340·19-s + (−1.20 + 1.32i)21-s + (−0.156 + 0.271i)23-s + (0.0898 + 0.155i)25-s + (−0.603 + 0.797i)27-s + (−0.648 − 1.12i)29-s + (−1.06 − 0.615i)31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.710 + 0.703i$
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: \((2,\ 1152,\ (\ :1/2),\ 0.710 + 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.573136408\)
\(L(\frac12)\) \(\approx\) \(1.573136408\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.65 - 0.524i)T \)
good5 \( 1 + (-1.01 + 1.75i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-4.09 + 2.36i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.93 - 1.69i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.51 - 3.18i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.87iT - 17T^{2} \)
19 \( 1 - 1.48T + 19T^{2} \)
23 \( 1 + (0.751 - 1.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.49 + 6.04i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.93 + 3.42i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.86iT - 37T^{2} \)
41 \( 1 + (-4.62 - 2.66i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.76 - 4.78i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.09 - 7.09i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 4.96T + 53T^{2} \)
59 \( 1 + (7.88 + 4.55i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.51 - 3.18i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.48 + 9.50i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.68T + 71T^{2} \)
73 \( 1 - 0.449T + 73T^{2} \)
79 \( 1 + (-5.93 + 3.42i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.06 + 3.50i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 0.142iT - 89T^{2} \)
97 \( 1 + (-0.724 - 1.25i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.615205817623803017454907373079, −9.126670106703510782032472957714, −7.84945781658673890466472965646, −7.36837286944502558714517398589, −6.11081670779403045354102673692, −5.28454085230537620770121256039, −4.64390606571161834559623696647, −3.94090952592795730889905703208, −1.86007286281715206881432669829, −0.922113514766020083477468826454, 1.33339783951642718632919514423, 2.36088166295061090778390965497, 3.74711066168139500478219946329, 5.22056013034253437047351017333, 5.59817191665945400193796060274, 6.30477825344495626019247289178, 7.45300311184326185091160352895, 8.228922716961726347018844283839, 8.850951734820128506580268867092, 10.49044936543857939989350461810

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