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Properties

Label 2-1152-16.11-c2-0-23
Degree $2$
Conductor $1152$
Sign $-0.209 + 0.977i$
Analytic cond. $31.3897$
Root an. cond. $5.60265$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.41 + 2.41i)5-s − 11.8·7-s + (−11.9 + 11.9i)11-s + (−2.08 + 2.08i)13-s + 23.1·17-s + (6.77 + 6.77i)19-s − 3.92·23-s − 13.3i·25-s + (0.782 − 0.782i)29-s + 2.65i·31-s + (−28.6 − 28.6i)35-s + (−37.2 − 37.2i)37-s − 69.1i·41-s + (29.2 − 29.2i)43-s − 68.0i·47-s + ⋯
L(s)  = 1  + (0.482 + 0.482i)5-s − 1.69·7-s + (−1.08 + 1.08i)11-s + (−0.160 + 0.160i)13-s + 1.36·17-s + (0.356 + 0.356i)19-s − 0.170·23-s − 0.534i·25-s + (0.0269 − 0.0269i)29-s + 0.0855i·31-s + (−0.818 − 0.818i)35-s + (−1.00 − 1.00i)37-s − 1.68i·41-s + (0.681 − 0.681i)43-s − 1.44i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.209 + 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.209 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.209 + 0.977i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ -0.209 + 0.977i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5678338369\)
\(L(\frac12)\) \(\approx\) \(0.5678338369\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (-2.41 - 2.41i)T + 25iT^{2} \)
7 \( 1 + 11.8T + 49T^{2} \)
11 \( 1 + (11.9 - 11.9i)T - 121iT^{2} \)
13 \( 1 + (2.08 - 2.08i)T - 169iT^{2} \)
17 \( 1 - 23.1T + 289T^{2} \)
19 \( 1 + (-6.77 - 6.77i)T + 361iT^{2} \)
23 \( 1 + 3.92T + 529T^{2} \)
29 \( 1 + (-0.782 + 0.782i)T - 841iT^{2} \)
31 \( 1 - 2.65iT - 961T^{2} \)
37 \( 1 + (37.2 + 37.2i)T + 1.36e3iT^{2} \)
41 \( 1 + 69.1iT - 1.68e3T^{2} \)
43 \( 1 + (-29.2 + 29.2i)T - 1.84e3iT^{2} \)
47 \( 1 + 68.0iT - 2.20e3T^{2} \)
53 \( 1 + (40.3 + 40.3i)T + 2.80e3iT^{2} \)
59 \( 1 + (-23.5 + 23.5i)T - 3.48e3iT^{2} \)
61 \( 1 + (65.9 - 65.9i)T - 3.72e3iT^{2} \)
67 \( 1 + (-40.9 - 40.9i)T + 4.48e3iT^{2} \)
71 \( 1 + 98.1T + 5.04e3T^{2} \)
73 \( 1 - 74.2iT - 5.32e3T^{2} \)
79 \( 1 - 0.779iT - 6.24e3T^{2} \)
83 \( 1 + (91.7 + 91.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 20.1iT - 7.92e3T^{2} \)
97 \( 1 + 86.6T + 9.40e3T^{2} \)
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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

−9.679417132747234020174736934400, −8.675748147818834373865010456444, −7.39405408315097891304206038565, −7.03633441045464763200615348773, −5.93269920231931209595758157554, −5.34256480479150799242146342131, −3.90515743184499973178626899818, −3.00774887220588264280575829078, −2.10114252487743554675268137252, −0.18412271300048277486473019582, 1.10133973568297715458304714959, 2.95222954024484059543274979749, 3.25443550255072638874329600573, 4.84041301848167053901310158416, 5.77650387888523213616699776415, 6.23312743035820180831336254277, 7.42620946008850765363128955535, 8.195263068750023436257696837040, 9.249202039820470010236343577672, 9.744568610678616773252159739306

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