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Properties

Label 2-1152-9.7-c1-0-45
Degree $2$
Conductor $1152$
Sign $-0.287 + 0.957i$
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  + (1.72 + 0.101i)3-s + (−1.24 − 2.15i)5-s + (0.909 − 1.57i)7-s + (2.97 + 0.350i)9-s + (0.598 − 1.03i)11-s + (−2.83 − 4.90i)13-s + (−1.93 − 3.84i)15-s − 5.30·17-s − 4.55·19-s + (1.73 − 2.63i)21-s + (−2.01 − 3.48i)23-s + (−0.589 + 1.02i)25-s + (5.11 + 0.909i)27-s + (−3.01 + 5.22i)29-s + (2.81 + 4.87i)31-s + ⋯
L(s)  = 1  + (0.998 + 0.0585i)3-s + (−0.555 − 0.962i)5-s + (0.343 − 0.595i)7-s + (0.993 + 0.116i)9-s + (0.180 − 0.312i)11-s + (−0.785 − 1.36i)13-s + (−0.498 − 0.993i)15-s − 1.28·17-s − 1.04·19-s + (0.377 − 0.574i)21-s + (−0.419 − 0.727i)23-s + (−0.117 + 0.204i)25-s + (0.984 + 0.174i)27-s + (−0.559 + 0.969i)29-s + (0.505 + 0.876i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.287 + 0.957i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.287 + 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.773161143\)
\(L(\frac12)\) \(\approx\) \(1.773161143\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.101i)T \)
good5 \( 1 + (1.24 + 2.15i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.909 + 1.57i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.598 + 1.03i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.83 + 4.90i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.30T + 17T^{2} \)
19 \( 1 + 4.55T + 19T^{2} \)
23 \( 1 + (2.01 + 3.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.01 - 5.22i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.81 - 4.87i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5.18T + 37T^{2} \)
41 \( 1 + (-4.57 - 7.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.99 + 6.91i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.39 + 2.41i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.54T + 53T^{2} \)
59 \( 1 + (1.85 + 3.21i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.01 + 6.95i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.91 + 11.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.1T + 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 + (-4.36 + 7.56i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.89 + 15.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.455T + 89T^{2} \)
97 \( 1 + (1.01 - 1.76i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.303099422338192081571306559150, −8.640445834707326751513002917321, −8.054755628670047813365354432511, −7.39255852619366808248729560830, −6.31181909119764397861383617272, −4.79669751382185208904069491923, −4.44146550438797347858454505818, −3.32841793648078109578457529473, −2.14980703466880795462637261607, −0.65719488674408006815123577991, 2.10063026408653145058470019787, 2.53586110975415561907995716339, 4.04623804272207889610852785697, 4.39671703762363778664966036866, 6.06621670048997680426224114746, 6.97367216810498069665544012211, 7.50390373580347253630819443344, 8.421135777467439638639693198317, 9.227736327444064579656206442384, 9.786186893795050427740081938441

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