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Properties

Label 2-1152-9.4-c1-0-36
Degree $2$
Conductor $1152$
Sign $0.558 + 0.829i$
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  + (1.69 − 0.359i)3-s + (1.74 − 3.01i)5-s + (1.34 + 2.33i)7-s + (2.74 − 1.21i)9-s + (−2.84 − 4.92i)11-s + (−1.76 + 3.04i)13-s + (1.86 − 5.74i)15-s + 7.65·17-s − 2.02·19-s + (3.12 + 3.47i)21-s + (0.0370 − 0.0642i)23-s + (−3.57 − 6.18i)25-s + (4.20 − 3.05i)27-s + (−2.46 − 4.26i)29-s + (−3.72 + 6.44i)31-s + ⋯
L(s)  = 1  + (0.978 − 0.207i)3-s + (0.779 − 1.34i)5-s + (0.510 + 0.883i)7-s + (0.913 − 0.406i)9-s + (−0.857 − 1.48i)11-s + (−0.488 + 0.845i)13-s + (0.481 − 1.48i)15-s + 1.85·17-s − 0.463·19-s + (0.682 + 0.758i)21-s + (0.00773 − 0.0133i)23-s + (−0.714 − 1.23i)25-s + (0.809 − 0.587i)27-s + (−0.456 − 0.791i)29-s + (−0.668 + 1.15i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.558 + 0.829i$
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: \((2,\ 1152,\ (\ :1/2),\ 0.558 + 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.728290346\)
\(L(\frac12)\) \(\approx\) \(2.728290346\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 + 0.359i)T \)
good5 \( 1 + (-1.74 + 3.01i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.34 - 2.33i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.84 + 4.92i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.76 - 3.04i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 7.65T + 17T^{2} \)
19 \( 1 + 2.02T + 19T^{2} \)
23 \( 1 + (-0.0370 + 0.0642i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.46 + 4.26i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.72 - 6.44i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.00T + 37T^{2} \)
41 \( 1 + (-0.482 + 0.834i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.255 + 0.442i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.83 - 4.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + (4.47 - 7.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.46 - 2.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.56 + 2.71i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.19T + 71T^{2} \)
73 \( 1 - 5.21T + 73T^{2} \)
79 \( 1 + (-0.716 - 1.24i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.74 - 3.01i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + (3.50 + 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.401906809733101964550144382749, −8.809516068193128353362158097938, −8.250589271984753809962070607441, −7.55366439122436537653455294978, −6.02837209144261047612610162969, −5.46238558486005548612849471973, −4.56720000170070762693888541227, −3.23267436560026477494899051563, −2.19246575884596333264589837945, −1.17985516401018560744408025489, 1.76279059036940818746508024389, 2.66877959698246510788697361968, 3.52166405244463634307869739559, 4.67598509472813234461727081694, 5.63569018564796773969376942739, 6.93094840174887211538183898680, 7.67903747037305849885897642649, 7.80985728950845321157766901192, 9.452477255772015311537037408158, 10.05358572956487360112516811850

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