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Properties

Label 2-1152-96.59-c1-0-4
Degree $2$
Conductor $1152$
Sign $-0.484 - 0.874i$
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  + (−0.366 + 0.885i)5-s + (−1.21 + 1.21i)7-s + (0.545 − 1.31i)11-s + (0.0270 − 0.0112i)13-s − 1.32·17-s + (1.73 + 4.19i)19-s + (0.934 − 0.934i)23-s + (2.88 + 2.88i)25-s + (−9.35 + 3.87i)29-s + 9.74i·31-s + (−0.630 − 1.52i)35-s + (−6.28 − 2.60i)37-s + (−3.42 − 3.42i)41-s + (−0.997 − 0.413i)43-s + 6.21i·47-s + ⋯
L(s)  = 1  + (−0.164 + 0.396i)5-s + (−0.459 + 0.459i)7-s + (0.164 − 0.396i)11-s + (0.00750 − 0.00310i)13-s − 0.321·17-s + (0.398 + 0.961i)19-s + (0.194 − 0.194i)23-s + (0.577 + 0.577i)25-s + (−1.73 + 0.719i)29-s + 1.75i·31-s + (−0.106 − 0.257i)35-s + (−1.03 − 0.427i)37-s + (−0.534 − 0.534i)41-s + (−0.152 − 0.0630i)43-s + 0.906i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $-0.484 - 0.874i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ -0.484 - 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9628598191\)
\(L(\frac12)\) \(\approx\) \(0.9628598191\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.366 - 0.885i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (1.21 - 1.21i)T - 7iT^{2} \)
11 \( 1 + (-0.545 + 1.31i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-0.0270 + 0.0112i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 1.32T + 17T^{2} \)
19 \( 1 + (-1.73 - 4.19i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (-0.934 + 0.934i)T - 23iT^{2} \)
29 \( 1 + (9.35 - 3.87i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 9.74iT - 31T^{2} \)
37 \( 1 + (6.28 + 2.60i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.42 + 3.42i)T + 41iT^{2} \)
43 \( 1 + (0.997 + 0.413i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 6.21iT - 47T^{2} \)
53 \( 1 + (2.94 + 1.22i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (10.4 + 4.32i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-2.76 - 6.68i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-10.1 + 4.18i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-7.38 - 7.38i)T + 71iT^{2} \)
73 \( 1 + (8.30 - 8.30i)T - 73iT^{2} \)
79 \( 1 + 5.54T + 79T^{2} \)
83 \( 1 + (11.4 - 4.74i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-7.93 + 7.93i)T - 89iT^{2} \)
97 \( 1 - 12.5T + 97T^{2} \)
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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

−10.10071047380236547824610045044, −9.131918019186815034948375482544, −8.612572902806488022370570987821, −7.46900032927459390583741885864, −6.80548806572077958695696073642, −5.84860173771994004512622988654, −5.07209334174922192700588327697, −3.68543058920167403378164904445, −3.04683058589861174212969703955, −1.61139477603347761612825575872, 0.41306020577384676167783903304, 2.01439847897602993746837308520, 3.34062908247258402907180083037, 4.29660929956336754580776448195, 5.14359246759111057622102539657, 6.24871186362380886750984919024, 7.07587684983512536605364692025, 7.80888612339031724229089921981, 8.824705927815270446986617250291, 9.515514066630729971821450171563

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