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Properties

Label 2-1152-48.35-c1-0-14
Degree $2$
Conductor $1152$
Sign $-0.643 + 0.765i$
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  + (0.763 + 0.763i)5-s − 1.33·7-s + (−1.95 + 1.95i)11-s + (−4.18 − 4.18i)13-s − 4.03i·17-s + (−4.26 + 4.26i)19-s − 8.86i·23-s − 3.83i·25-s + (1.23 − 1.23i)29-s + 2.87i·31-s + (−1.02 − 1.02i)35-s + (−0.434 + 0.434i)37-s + 7.81·41-s + (−5.49 − 5.49i)43-s − 3.20·47-s + ⋯
L(s)  = 1  + (0.341 + 0.341i)5-s − 0.505·7-s + (−0.590 + 0.590i)11-s + (−1.16 − 1.16i)13-s − 0.978i·17-s + (−0.979 + 0.979i)19-s − 1.84i·23-s − 0.766i·25-s + (0.230 − 0.230i)29-s + 0.516i·31-s + (−0.172 − 0.172i)35-s + (−0.0714 + 0.0714i)37-s + 1.21·41-s + (−0.838 − 0.838i)43-s − 0.467·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.643 + 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.643 + 0.765i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.643 + 0.765i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5785465493\)
\(L(\frac12)\) \(\approx\) \(0.5785465493\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.763 - 0.763i)T + 5iT^{2} \)
7 \( 1 + 1.33T + 7T^{2} \)
11 \( 1 + (1.95 - 1.95i)T - 11iT^{2} \)
13 \( 1 + (4.18 + 4.18i)T + 13iT^{2} \)
17 \( 1 + 4.03iT - 17T^{2} \)
19 \( 1 + (4.26 - 4.26i)T - 19iT^{2} \)
23 \( 1 + 8.86iT - 23T^{2} \)
29 \( 1 + (-1.23 + 1.23i)T - 29iT^{2} \)
31 \( 1 - 2.87iT - 31T^{2} \)
37 \( 1 + (0.434 - 0.434i)T - 37iT^{2} \)
41 \( 1 - 7.81T + 41T^{2} \)
43 \( 1 + (5.49 + 5.49i)T + 43iT^{2} \)
47 \( 1 + 3.20T + 47T^{2} \)
53 \( 1 + (-4.06 - 4.06i)T + 53iT^{2} \)
59 \( 1 + (-4.71 + 4.71i)T - 59iT^{2} \)
61 \( 1 + (3.26 + 3.26i)T + 61iT^{2} \)
67 \( 1 + (5.44 - 5.44i)T - 67iT^{2} \)
71 \( 1 - 3.76iT - 71T^{2} \)
73 \( 1 + 10.5iT - 73T^{2} \)
79 \( 1 - 11.1iT - 79T^{2} \)
83 \( 1 + (9.73 + 9.73i)T + 83iT^{2} \)
89 \( 1 - 1.64T + 89T^{2} \)
97 \( 1 + 5.70T + 97T^{2} \)
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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

−9.838540563858841296180598102338, −8.618943144879887729575727552848, −7.84683618833459127085557003475, −6.96791131532960356559980067774, −6.20991472195774924048051140533, −5.19865485367771628547693314474, −4.37107781746088551933890013044, −2.92238332407387129417875533483, −2.31121245967201266160076259395, −0.23196881653840357469760735910, 1.68794411594019853430732158063, 2.82966494234628033509765243309, 4.01648424919913170631923825865, 5.00491747680908749722759325571, 5.86656479312152912737882554940, 6.75975010797860538157034957097, 7.58365691138225985328433160885, 8.576546797336235452584544070606, 9.382378013401124241265950444865, 9.864686102145832343621082076707

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