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Properties

Label 2-1152-72.13-c1-0-7
Degree $2$
Conductor $1152$
Sign $0.467 - 0.884i$
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  + (−1.29 − 1.14i)3-s + (2.41 + 1.39i)5-s + (−0.551 − 0.955i)7-s + (0.372 + 2.97i)9-s + (−2.58 + 1.49i)11-s + (0.220 + 0.127i)13-s + (−1.53 − 4.56i)15-s + 3.78·17-s + 6.46i·19-s + (−0.378 + 1.87i)21-s + (−2.63 + 4.56i)23-s + (1.37 + 2.37i)25-s + (2.92 − 4.29i)27-s + (−8.06 + 4.65i)29-s + (3.74 − 6.48i)31-s + ⋯
L(s)  = 1  + (−0.749 − 0.661i)3-s + (1.07 + 0.622i)5-s + (−0.208 − 0.361i)7-s + (0.124 + 0.992i)9-s + (−0.779 + 0.449i)11-s + (0.0611 + 0.0352i)13-s + (−0.396 − 1.17i)15-s + 0.917·17-s + 1.48i·19-s + (−0.0826 + 0.408i)21-s + (−0.550 + 0.952i)23-s + (0.274 + 0.475i)25-s + (0.563 − 0.826i)27-s + (−1.49 + 0.864i)29-s + (0.671 − 1.16i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.467 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.467 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.467 - 0.884i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.467 - 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.183389699\)
\(L(\frac12)\) \(\approx\) \(1.183389699\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.29 + 1.14i)T \)
good5 \( 1 + (-2.41 - 1.39i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.551 + 0.955i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.58 - 1.49i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.220 - 0.127i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 3.78T + 17T^{2} \)
19 \( 1 - 6.46iT - 19T^{2} \)
23 \( 1 + (2.63 - 4.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (8.06 - 4.65i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.74 + 6.48i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2.52iT - 37T^{2} \)
41 \( 1 + (0.764 - 1.32i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.85 - 2.80i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.13 - 8.89i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.962iT - 53T^{2} \)
59 \( 1 + (-6.76 - 3.90i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.5 - 6.65i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.55 - 3.78i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.17T + 71T^{2} \)
73 \( 1 - 5.34T + 73T^{2} \)
79 \( 1 + (1.94 + 3.37i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-14.4 + 8.32i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 10.0T + 89T^{2} \)
97 \( 1 + (-7.54 - 13.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.21832496417515906211616705707, −9.412575578353755996722480604178, −7.83916147003592070515738913929, −7.55591324199400252681429041957, −6.41593284275000896944192176663, −5.84100065796444211055638630417, −5.16836616534243124429398689564, −3.71322628614647766380439926492, −2.38740454386056494619056099109, −1.44081487469969825869079877530, 0.57530198415629867339730459103, 2.22894573703588725631176410995, 3.47247830686355857120821443107, 4.78725248428363296384545518813, 5.40573386416017319165781832931, 5.99106056705107693315034648974, 6.92453743421844151140091211214, 8.237338799225897926604033370530, 9.098070171514408271707909471402, 9.657627530909927370596513226799

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