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Properties

Label 2-1152-72.11-c1-0-32
Degree $2$
Conductor $1152$
Sign $0.500 + 0.865i$
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  + (−0.0745 + 1.73i)3-s + (−2.11 − 3.66i)5-s + (3.25 + 1.87i)7-s + (−2.98 − 0.258i)9-s + (0.424 + 0.245i)11-s + (1.78 − 1.03i)13-s + (6.49 − 3.38i)15-s − 4.97i·17-s − 6.32·19-s + (−3.49 + 5.49i)21-s + (−0.0297 − 0.0514i)23-s + (−6.44 + 11.1i)25-s + (0.669 − 5.15i)27-s + (2.80 − 4.86i)29-s + (0.546 − 0.315i)31-s + ⋯
L(s)  = 1  + (−0.0430 + 0.999i)3-s + (−0.946 − 1.63i)5-s + (1.23 + 0.710i)7-s + (−0.996 − 0.0860i)9-s + (0.128 + 0.0739i)11-s + (0.496 − 0.286i)13-s + (1.67 − 0.874i)15-s − 1.20i·17-s − 1.45·19-s + (−0.762 + 1.19i)21-s + (−0.00619 − 0.0107i)23-s + (−1.28 + 2.23i)25-s + (0.128 − 0.991i)27-s + (0.521 − 0.903i)29-s + (0.0981 − 0.0566i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.500 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.500 + 0.865i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.500 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.202878907\)
\(L(\frac12)\) \(\approx\) \(1.202878907\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0745 - 1.73i)T \)
good5 \( 1 + (2.11 + 3.66i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-3.25 - 1.87i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.424 - 0.245i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.78 + 1.03i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.97iT - 17T^{2} \)
19 \( 1 + 6.32T + 19T^{2} \)
23 \( 1 + (0.0297 + 0.0514i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.80 + 4.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.546 + 0.315i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 6.74iT - 37T^{2} \)
41 \( 1 + (-4.44 + 2.56i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.35 + 7.53i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.58 + 7.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 1.32T + 53T^{2} \)
59 \( 1 + (6.67 - 3.85i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.56 - 1.48i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.07 + 3.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.06T + 71T^{2} \)
73 \( 1 - 8.03T + 73T^{2} \)
79 \( 1 + (0.601 + 0.347i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.46 - 4.88i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 6.19iT - 89T^{2} \)
97 \( 1 + (0.864 - 1.49i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.330843848213808092835326327777, −8.794832039629473811594663698951, −8.350375477216174765998497373629, −7.55193173281981447881054613942, −5.85897311439689520285665721641, −5.14290763741408268970746808376, −4.49605787464431542175602945451, −3.87372150809158241943468472652, −2.24009310005877568690144477921, −0.56372902624290302794932239373, 1.39720409829348193729663215506, 2.59345608010481736902527152218, 3.71175970266137912601159098889, 4.56333149863109324799158563193, 6.27810279456471564512012427762, 6.53077734087791239222168966051, 7.62239771080371433967660715511, 7.948795526516592345766996811764, 8.720591585270550551187581844027, 10.40673075549759167160492074614

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