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Properties

Label 2-1152-32.13-c1-0-9
Degree $2$
Conductor $1152$
Sign $-0.171 + 0.985i$
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  + (−2.42 + 1.00i)5-s + (−3.40 + 3.40i)7-s + (0.847 + 2.04i)11-s + (−1.82 − 0.757i)13-s − 7.04i·17-s + (4.96 + 2.05i)19-s + (−3.37 − 3.37i)23-s + (1.32 − 1.32i)25-s + (2.83 − 6.84i)29-s − 1.94·31-s + (4.83 − 11.6i)35-s + (4.02 − 1.66i)37-s + (−0.970 − 0.970i)41-s + (0.0467 + 0.112i)43-s − 6.49i·47-s + ⋯
L(s)  = 1  + (−1.08 + 0.448i)5-s + (−1.28 + 1.28i)7-s + (0.255 + 0.616i)11-s + (−0.507 − 0.210i)13-s − 1.70i·17-s + (1.14 + 0.472i)19-s + (−0.703 − 0.703i)23-s + (0.265 − 0.265i)25-s + (0.526 − 1.27i)29-s − 0.348·31-s + (0.817 − 1.97i)35-s + (0.661 − 0.274i)37-s + (−0.151 − 0.151i)41-s + (0.00712 + 0.0172i)43-s − 0.946i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.171 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.171 + 0.985i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.171 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3305968346\)
\(L(\frac12)\) \(\approx\) \(0.3305968346\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (2.42 - 1.00i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (3.40 - 3.40i)T - 7iT^{2} \)
11 \( 1 + (-0.847 - 2.04i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.82 + 0.757i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 7.04iT - 17T^{2} \)
19 \( 1 + (-4.96 - 2.05i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (3.37 + 3.37i)T + 23iT^{2} \)
29 \( 1 + (-2.83 + 6.84i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 1.94T + 31T^{2} \)
37 \( 1 + (-4.02 + 1.66i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.970 + 0.970i)T + 41iT^{2} \)
43 \( 1 + (-0.0467 - 0.112i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 6.49iT - 47T^{2} \)
53 \( 1 + (-0.894 - 2.15i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (7.71 - 3.19i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (4.20 - 10.1i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (0.933 - 2.25i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (-4.81 + 4.81i)T - 71iT^{2} \)
73 \( 1 + (2.31 + 2.31i)T + 73iT^{2} \)
79 \( 1 + 6.41iT - 79T^{2} \)
83 \( 1 + (6.76 + 2.80i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (8.37 - 8.37i)T - 89iT^{2} \)
97 \( 1 + 5.80T + 97T^{2} \)
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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

−9.618148603083933757870901966232, −8.881438228299002308126139120317, −7.74821909700373921966170481751, −7.18789805844064499005130441996, −6.26244383720660826143922967022, −5.36152747807929487804740078430, −4.23385718029115545534440390021, −3.13886182074275266895507248683, −2.49834843395888450926418408958, −0.16270000699166938139732972910, 1.15795414075531951687755832576, 3.24736427980775406082939502226, 3.76574067161382498059343701704, 4.62047461825668376466824112934, 5.94909727404852333444340442741, 6.79994959468613652342231827956, 7.57556041635612966156379755296, 8.253692251814379947365137092529, 9.289852821006928237099730840571, 9.986314740791134887772629350119

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