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Properties

Label 2-1152-96.83-c1-0-3
Degree $2$
Conductor $1152$
Sign $-0.124 - 0.992i$
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  + (1.39 + 3.36i)5-s + (1.05 + 1.05i)7-s + (1.50 + 3.64i)11-s + (−2.23 − 0.927i)13-s + 7.49·17-s + (−0.818 + 1.97i)19-s + (−5.80 − 5.80i)23-s + (−5.86 + 5.86i)25-s + (−0.326 − 0.135i)29-s + 1.71i·31-s + (−2.08 + 5.03i)35-s + (0.387 − 0.160i)37-s + (1.50 − 1.50i)41-s + (−7.40 + 3.06i)43-s + 7.27i·47-s + ⋯
L(s)  = 1  + (0.624 + 1.50i)5-s + (0.399 + 0.399i)7-s + (0.455 + 1.09i)11-s + (−0.620 − 0.257i)13-s + 1.81·17-s + (−0.187 + 0.453i)19-s + (−1.21 − 1.21i)23-s + (−1.17 + 1.17i)25-s + (−0.0606 − 0.0251i)29-s + 0.307i·31-s + (−0.352 + 0.851i)35-s + (0.0636 − 0.0263i)37-s + (0.234 − 0.234i)41-s + (−1.12 + 0.467i)43-s + 1.06i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.124 - 0.992i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.124 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.840023350\)
\(L(\frac12)\) \(\approx\) \(1.840023350\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.39 - 3.36i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (-1.05 - 1.05i)T + 7iT^{2} \)
11 \( 1 + (-1.50 - 3.64i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (2.23 + 0.927i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 - 7.49T + 17T^{2} \)
19 \( 1 + (0.818 - 1.97i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (5.80 + 5.80i)T + 23iT^{2} \)
29 \( 1 + (0.326 + 0.135i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 1.71iT - 31T^{2} \)
37 \( 1 + (-0.387 + 0.160i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-1.50 + 1.50i)T - 41iT^{2} \)
43 \( 1 + (7.40 - 3.06i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 7.27iT - 47T^{2} \)
53 \( 1 + (3.94 - 1.63i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-12.7 + 5.30i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (0.579 - 1.40i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-7.96 - 3.29i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-4.75 + 4.75i)T - 71iT^{2} \)
73 \( 1 + (7.99 + 7.99i)T + 73iT^{2} \)
79 \( 1 + 14.3T + 79T^{2} \)
83 \( 1 + (-1.35 - 0.560i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (-4.75 - 4.75i)T + 89iT^{2} \)
97 \( 1 + 1.28T + 97T^{2} \)
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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

−10.09984490046391261106805444610, −9.537764642497345835864631114191, −8.194686308311842752578151434888, −7.50192119538447002158049581312, −6.65940251425015810703659182842, −5.93959197182704632788408783078, −4.97958049966916920897478121750, −3.72522352999752796687323841596, −2.66604584951215502367556389529, −1.81101598466598897252212571991, 0.834144628817547627703111562237, 1.80565684647212742567882209126, 3.44892658402590737012168947276, 4.44459021102277395765339032830, 5.44178889603353542814377328252, 5.84328486709463524732644862613, 7.20997811898312692462160752285, 8.152941902142723442596979738481, 8.672282780818935668211594901667, 9.709691947830351074110494204188

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