Properties

Label 2-1152-48.11-c1-0-3
Degree $2$
Conductor $1152$
Sign $-0.351 - 0.936i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.351 - 0.936i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.351 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9992437442\)
\(L(\frac12)\) \(\approx\) \(0.9992437442\)
\(L(\frac{3}{2})\) 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

−9.864010684409221063300311332557, −9.413843514851316144714445339052, −8.308873933845171591757322617280, −7.41061353487868000477387627939, −7.07996656646952993046829877247, −5.60413036528285178583744610605, −5.05965771545045169108669779309, −3.84106097670396870949317590115, −2.89193578401666380586860720634, −1.58226464185772586945224340058, 0.42769768670912195208352153951, 2.14346257923177561722377646338, 3.17068063529767704388523370524, 4.68622913977101672689098319580, 4.91913277374457810474400451757, 6.16328904967932361721787813284, 7.24022468974718527231792374264, 7.972835094962028674388608849658, 8.510573617483947363617159980854, 9.685362538800088505006037231414

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