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Properties

Label 2-1152-48.35-c1-0-3
Degree $2$
Conductor $1152$
Sign $-0.555 - 0.831i$
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  + (2.10 + 2.10i)5-s − 4.40·7-s + (−0.215 + 0.215i)11-s + (2.73 + 2.73i)13-s − 2.36i·17-s + (−0.758 + 0.758i)19-s + 1.75i·23-s + 3.86i·25-s + (−5.54 + 5.54i)29-s + 9.01i·31-s + (−9.27 − 9.27i)35-s + (−3.10 + 3.10i)37-s − 10.1·41-s + (3.54 + 3.54i)43-s − 3.90·47-s + ⋯
L(s)  = 1  + (0.941 + 0.941i)5-s − 1.66·7-s + (−0.0650 + 0.0650i)11-s + (0.758 + 0.758i)13-s − 0.573i·17-s + (−0.174 + 0.174i)19-s + 0.366i·23-s + 0.772i·25-s + (−1.02 + 1.02i)29-s + 1.61i·31-s + (−1.56 − 1.56i)35-s + (−0.510 + 0.510i)37-s − 1.57·41-s + (0.540 + 0.540i)43-s − 0.569·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.555 - 0.831i$
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: \((2,\ 1152,\ (\ :1/2),\ -0.555 - 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.122661774\)
\(L(\frac12)\) \(\approx\) \(1.122661774\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.10 - 2.10i)T + 5iT^{2} \)
7 \( 1 + 4.40T + 7T^{2} \)
11 \( 1 + (0.215 - 0.215i)T - 11iT^{2} \)
13 \( 1 + (-2.73 - 2.73i)T + 13iT^{2} \)
17 \( 1 + 2.36iT - 17T^{2} \)
19 \( 1 + (0.758 - 0.758i)T - 19iT^{2} \)
23 \( 1 - 1.75iT - 23T^{2} \)
29 \( 1 + (5.54 - 5.54i)T - 29iT^{2} \)
31 \( 1 - 9.01iT - 31T^{2} \)
37 \( 1 + (3.10 - 3.10i)T - 37iT^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 + (-3.54 - 3.54i)T + 43iT^{2} \)
47 \( 1 + 3.90T + 47T^{2} \)
53 \( 1 + (2.71 + 2.71i)T + 53iT^{2} \)
59 \( 1 + (-3.40 + 3.40i)T - 59iT^{2} \)
61 \( 1 + (-1.75 - 1.75i)T + 61iT^{2} \)
67 \( 1 + (9.11 - 9.11i)T - 67iT^{2} \)
71 \( 1 + 11.8iT - 71T^{2} \)
73 \( 1 + 0.482iT - 73T^{2} \)
79 \( 1 + 6.88iT - 79T^{2} \)
83 \( 1 + (-4.79 - 4.79i)T + 83iT^{2} \)
89 \( 1 - 7.00T + 89T^{2} \)
97 \( 1 + 3.34T + 97T^{2} \)
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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

−10.09485001154987906356692399547, −9.361562431633284496121825872406, −8.740201890205483923605962814234, −7.24474630723900393775883521379, −6.62893332442615191516295786533, −6.17051464591963367587495103069, −5.11892382785761999860929963260, −3.56077201848078925373585175151, −3.03824875035470856911042887812, −1.75652428524407502151879260205, 0.46270771844146929070686769492, 2.00302103902853369723258152299, 3.24657314844244105226437185102, 4.19023464271349272471872223540, 5.60842163338387549158306975938, 5.92704674014000030797099159567, 6.82156708021606506871920557091, 8.035601805200592488791599626460, 8.880420708801386176689047524586, 9.546767039798437104112602625172

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