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Properties

Label 2-1152-9.4-c1-0-21
Degree $2$
Conductor $1152$
Sign $0.758 - 0.651i$
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  + (−0.857 + 1.50i)3-s + (0.551 − 0.955i)5-s + (1.62 + 2.81i)7-s + (−1.53 − 2.58i)9-s + (−1.28 − 2.23i)11-s + (1.58 − 2.74i)13-s + (0.965 + 1.64i)15-s + 4.71·17-s + 5.75·19-s + (−5.62 + 0.0333i)21-s + (−2.35 + 4.07i)23-s + (1.89 + 3.27i)25-s + (5.19 − 0.0922i)27-s + (−3.66 − 6.34i)29-s + (2.93 − 5.07i)31-s + ⋯
L(s)  = 1  + (−0.494 + 0.868i)3-s + (0.246 − 0.427i)5-s + (0.614 + 1.06i)7-s + (−0.510 − 0.860i)9-s + (−0.388 − 0.672i)11-s + (0.440 − 0.762i)13-s + (0.249 + 0.425i)15-s + 1.14·17-s + 1.32·19-s + (−1.22 + 0.00726i)21-s + (−0.490 + 0.850i)23-s + (0.378 + 0.655i)25-s + (0.999 − 0.0177i)27-s + (−0.680 − 1.17i)29-s + (0.526 − 0.911i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.758 - 0.651i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.758 - 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.582755335\)
\(L(\frac12)\) \(\approx\) \(1.582755335\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.857 - 1.50i)T \)
good5 \( 1 + (-0.551 + 0.955i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.62 - 2.81i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.28 + 2.23i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.58 + 2.74i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.71T + 17T^{2} \)
19 \( 1 - 5.75T + 19T^{2} \)
23 \( 1 + (2.35 - 4.07i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.66 + 6.34i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.93 + 5.07i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.0714T + 37T^{2} \)
41 \( 1 + (1.63 - 2.83i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.12 - 3.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.72 - 8.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.42T + 53T^{2} \)
59 \( 1 + (-4.19 + 7.26i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.66 - 8.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.09 - 10.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.335T + 71T^{2} \)
73 \( 1 - 14.8T + 73T^{2} \)
79 \( 1 + (4.85 + 8.41i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.07 - 5.31i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.42T + 89T^{2} \)
97 \( 1 + (-6.39 - 11.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.695652130321715634513390133371, −9.337139227519848750508450717812, −8.263129618901701993430125197706, −7.72281665681534980947387204837, −5.97526497773171805378628785159, −5.61242659638639199381055061815, −5.03330185659460191781940961009, −3.72216939497013909724984616415, −2.79640038969377588265469991046, −1.06951400179234245973502340704, 1.00527602144814151852162279311, 2.04769117151604917624709352055, 3.41617349053695515815707650961, 4.69255553926721215590128519769, 5.46731924929456300378863684292, 6.58150135859643197367081970481, 7.20461437889369206175448174265, 7.77312436904818240029914012855, 8.720177259406824935424126518560, 10.00635511030527689731522099087

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