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Properties

Label 2-1152-96.11-c1-0-9
Degree $2$
Conductor $1152$
Sign $0.0714 + 0.997i$
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  + (−1.64 − 0.682i)5-s + (−2.51 + 2.51i)7-s + (2.69 + 1.11i)11-s + (−1.76 − 4.25i)13-s + 6.10·17-s + (−3.43 + 1.42i)19-s + (0.525 − 0.525i)23-s + (−1.28 − 1.28i)25-s + (−1.46 − 3.53i)29-s − 7.55i·31-s + (5.85 − 2.42i)35-s + (2.30 − 5.56i)37-s + (−3.04 − 3.04i)41-s + (3.31 − 8.00i)43-s + 8.59i·47-s + ⋯
L(s)  = 1  + (−0.737 − 0.305i)5-s + (−0.949 + 0.949i)7-s + (0.811 + 0.336i)11-s + (−0.488 − 1.17i)13-s + 1.47·17-s + (−0.787 + 0.326i)19-s + (0.109 − 0.109i)23-s + (−0.256 − 0.256i)25-s + (−0.272 − 0.656i)29-s − 1.35i·31-s + (0.990 − 0.410i)35-s + (0.378 − 0.914i)37-s + (−0.475 − 0.475i)41-s + (0.505 − 1.22i)43-s + 1.25i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0714 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0714 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $0.0714 + 0.997i$
Analytic conductor: \(9.19876\)
Root analytic conductor: \(3.03294\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1/2),\ 0.0714 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8873253519\)
\(L(\frac12)\) \(\approx\) \(0.8873253519\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 + 0.682i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (2.51 - 2.51i)T - 7iT^{2} \)
11 \( 1 + (-2.69 - 1.11i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + (1.76 + 4.25i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 - 6.10T + 17T^{2} \)
19 \( 1 + (3.43 - 1.42i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (-0.525 + 0.525i)T - 23iT^{2} \)
29 \( 1 + (1.46 + 3.53i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + 7.55iT - 31T^{2} \)
37 \( 1 + (-2.30 + 5.56i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (3.04 + 3.04i)T + 41iT^{2} \)
43 \( 1 + (-3.31 + 8.00i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 - 8.59iT - 47T^{2} \)
53 \( 1 + (-3.78 + 9.14i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-3.73 + 9.01i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-3.41 + 1.41i)T + (43.1 - 43.1i)T^{2} \)
67 \( 1 + (0.0538 + 0.130i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (-1.31 - 1.31i)T + 71iT^{2} \)
73 \( 1 + (10.9 - 10.9i)T - 73iT^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + (4.38 + 10.5i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (-5.97 + 5.97i)T - 89iT^{2} \)
97 \( 1 + 3.21T + 97T^{2} \)
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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

−9.710213603922755587922395536259, −8.751840103127507823261867683790, −7.989177333022170897677476699080, −7.24002099800714221264625047795, −6.04660775216110990381065691437, −5.54699459016920727023693582597, −4.21603224947466883415849945428, −3.40987331749989253797104048142, −2.29133559636311085537713327620, −0.42606718885155916009956629086, 1.26054484980649349158293814768, 3.07176194334607243151054133345, 3.77195868228701505522400450560, 4.58455527814751228151237772010, 5.95669810835031251985950318484, 6.95100700367519089032324824829, 7.20239664206884080356769626288, 8.356083066295375227226847640568, 9.262195494618314486271441745189, 9.992806606592788485570341967180

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