Properties

Label 2-1152-72.59-c1-0-26
Degree $2$
Conductor $1152$
Sign $0.981 + 0.191i$
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  + (−0.944 + 1.45i)3-s + (−0.637 + 1.10i)5-s + (1.73 − 1.00i)7-s + (−1.21 − 2.74i)9-s + (0.511 − 0.295i)11-s + (−2.70 − 1.56i)13-s + (−1.00 − 1.96i)15-s − 4.34i·17-s + 4.36·19-s + (−0.183 + 3.46i)21-s + (4.02 − 6.97i)23-s + (1.68 + 2.92i)25-s + (5.13 + 0.821i)27-s + (−3.39 − 5.87i)29-s + (−0.688 − 0.397i)31-s + ⋯
L(s)  = 1  + (−0.545 + 0.838i)3-s + (−0.285 + 0.493i)5-s + (0.655 − 0.378i)7-s + (−0.405 − 0.913i)9-s + (0.154 − 0.0891i)11-s + (−0.750 − 0.433i)13-s + (−0.258 − 0.508i)15-s − 1.05i·17-s + 1.00·19-s + (−0.0400 + 0.756i)21-s + (0.839 − 1.45i)23-s + (0.337 + 0.584i)25-s + (0.987 + 0.158i)27-s + (−0.629 − 1.09i)29-s + (−0.123 − 0.0713i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.981 + 0.191i$
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: \((2,\ 1152,\ (\ :1/2),\ 0.981 + 0.191i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.220531528\)
\(L(\frac12)\) \(\approx\) \(1.220531528\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.944 - 1.45i)T \)
good5 \( 1 + (0.637 - 1.10i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.73 + 1.00i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.511 + 0.295i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.70 + 1.56i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.34iT - 17T^{2} \)
19 \( 1 - 4.36T + 19T^{2} \)
23 \( 1 + (-4.02 + 6.97i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.39 + 5.87i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.688 + 0.397i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.791iT - 37T^{2} \)
41 \( 1 + (3.81 + 2.20i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.84 - 4.93i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.881 - 1.52i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.01T + 53T^{2} \)
59 \( 1 + (-6.66 - 3.84i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-11.9 + 6.90i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.968 + 1.67i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.859T + 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 + (-14.6 + 8.43i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.89 + 3.40i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.00iT - 89T^{2} \)
97 \( 1 + (-5.03 - 8.71i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.775268960207290380665657571752, −9.231137914178265593842772106992, −8.054357838818208784378106099368, −7.27083689507236420956349735973, −6.44240659575975255820979364505, −5.21442201278630982729619310370, −4.76826647130229833379520416960, −3.63879886330917279010022465239, −2.68122592337281746925428895155, −0.67450544366695134808094460115, 1.17871245134030567290490751132, 2.16022269672046176718377690021, 3.66222438220490972827590385774, 5.08201289996657682756217923677, 5.32157230568219356556255686893, 6.60161237277939522613992113218, 7.34619984593697644883510793393, 8.092866581403049755718052392602, 8.846214921407233643136016892238, 9.771161053553807487575403516353

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