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Properties

Label 2-1152-12.11-c3-0-19
Degree $2$
Conductor $1152$
Sign $0.816 - 0.577i$
Analytic cond. $67.9702$
Root an. cond. $8.24440$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.77i·5-s − 16.6i·7-s − 12.3·11-s + 74.0·13-s − 18.4i·17-s + 106. i·19-s − 116.·23-s + 29.4·25-s − 93.6i·29-s + 158. i·31-s + 162.·35-s + 50.7·37-s − 315. i·41-s − 254. i·43-s − 355.·47-s + ⋯
L(s)  = 1  + 0.874i·5-s − 0.896i·7-s − 0.338·11-s + 1.57·13-s − 0.262i·17-s + 1.28i·19-s − 1.05·23-s + 0.235·25-s − 0.599i·29-s + 0.916i·31-s + 0.784·35-s + 0.225·37-s − 1.20i·41-s − 0.903i·43-s − 1.10·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.816 - 0.577i$
Analytic conductor: \(67.9702\)
Root analytic conductor: \(8.24440\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :3/2),\ 0.816 - 0.577i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.101434465\)
\(L(\frac12)\) \(\approx\) \(2.101434465\)
\(L(\frac{5}{2})\) 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 \( 1 - 9.77iT - 125T^{2} \)
7 \( 1 + 16.6iT - 343T^{2} \)
11 \( 1 + 12.3T + 1.33e3T^{2} \)
13 \( 1 - 74.0T + 2.19e3T^{2} \)
17 \( 1 + 18.4iT - 4.91e3T^{2} \)
19 \( 1 - 106. iT - 6.85e3T^{2} \)
23 \( 1 + 116.T + 1.21e4T^{2} \)
29 \( 1 + 93.6iT - 2.43e4T^{2} \)
31 \( 1 - 158. iT - 2.97e4T^{2} \)
37 \( 1 - 50.7T + 5.06e4T^{2} \)
41 \( 1 + 315. iT - 6.89e4T^{2} \)
43 \( 1 + 254. iT - 7.95e4T^{2} \)
47 \( 1 + 355.T + 1.03e5T^{2} \)
53 \( 1 - 150. iT - 1.48e5T^{2} \)
59 \( 1 - 504.T + 2.05e5T^{2} \)
61 \( 1 - 281.T + 2.26e5T^{2} \)
67 \( 1 + 101. iT - 3.00e5T^{2} \)
71 \( 1 - 797.T + 3.57e5T^{2} \)
73 \( 1 - 1.00e3T + 3.89e5T^{2} \)
79 \( 1 - 189. iT - 4.93e5T^{2} \)
83 \( 1 + 475.T + 5.71e5T^{2} \)
89 \( 1 - 1.52e3iT - 7.04e5T^{2} \)
97 \( 1 - 700.T + 9.12e5T^{2} \)
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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

−9.700659583078965074485101677445, −8.501206198310666941873567143867, −7.88214424640161828076930385064, −6.93823434067807208934492416293, −6.28727392530114298182842749003, −5.35753330915335745791999856884, −3.92613720059093326079185016995, −3.52583735655678375933260974395, −2.14603317122278167024385987889, −0.876419348344388113585903907126, 0.66867542158694727101592620528, 1.84084583119309565337861685554, 3.02728488983373063779265158073, 4.19317671211415101079716278478, 5.10797618753897388396651936318, 5.90456904979894808725034240049, 6.67317875183414525446911084362, 8.041806844418248117699285033420, 8.515352574074020633847258322359, 9.189085454639054682787450644286

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