Properties

Label 2-1150-23.22-c2-0-18
Degree $2$
Conductor $1150$
Sign $0.756 - 0.653i$
Analytic cond. $31.3352$
Root an. cond. $5.59778$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s − 1.43·3-s + 2.00·4-s + 2.03·6-s + 10.1i·7-s − 2.82·8-s − 6.93·9-s − 13.0i·11-s − 2.87·12-s − 23.2·13-s − 14.4i·14-s + 4.00·16-s − 28.2i·17-s + 9.80·18-s + 11.6i·19-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.479·3-s + 0.500·4-s + 0.339·6-s + 1.45i·7-s − 0.353·8-s − 0.770·9-s − 1.18i·11-s − 0.239·12-s − 1.78·13-s − 1.02i·14-s + 0.250·16-s − 1.66i·17-s + 0.544·18-s + 0.614i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $0.756 - 0.653i$
Analytic conductor: \(31.3352\)
Root analytic conductor: \(5.59778\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1150} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1150,\ (\ :1),\ 0.756 - 0.653i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6928127694\)
\(L(\frac12)\) \(\approx\) \(0.6928127694\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
5 \( 1 \)
23 \( 1 + (-17.3 + 15.0i)T \)
good3 \( 1 + 1.43T + 9T^{2} \)
7 \( 1 - 10.1iT - 49T^{2} \)
11 \( 1 + 13.0iT - 121T^{2} \)
13 \( 1 + 23.2T + 169T^{2} \)
17 \( 1 + 28.2iT - 289T^{2} \)
19 \( 1 - 11.6iT - 361T^{2} \)
29 \( 1 - 42.4T + 841T^{2} \)
31 \( 1 - 18.7T + 961T^{2} \)
37 \( 1 + 1.14iT - 1.36e3T^{2} \)
41 \( 1 + 72.8T + 1.68e3T^{2} \)
43 \( 1 + 4.96iT - 1.84e3T^{2} \)
47 \( 1 - 0.813T + 2.20e3T^{2} \)
53 \( 1 - 26.7iT - 2.80e3T^{2} \)
59 \( 1 - 94.0T + 3.48e3T^{2} \)
61 \( 1 - 74.5iT - 3.72e3T^{2} \)
67 \( 1 - 80.0iT - 4.48e3T^{2} \)
71 \( 1 + 83.5T + 5.04e3T^{2} \)
73 \( 1 - 8.98T + 5.32e3T^{2} \)
79 \( 1 - 80.2iT - 6.24e3T^{2} \)
83 \( 1 + 94.6iT - 6.88e3T^{2} \)
89 \( 1 - 136. iT - 7.92e3T^{2} \)
97 \( 1 + 2.32iT - 9.40e3T^{2} \)
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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

−9.659738527650348556690971192020, −8.715855879601780764660477650328, −8.430506195451149380692040504578, −7.21047796711811875764842658187, −6.36981198053291401391835802769, −5.45008556647562468321683193266, −4.95301989305221993796954575723, −2.85471315899153266499965500928, −2.60074433682146473974356715496, −0.65975677197627460369577936732, 0.46167230019885941694539316298, 1.82595263521715727628555735477, 3.10799324280939369565542974702, 4.45282590665044072242909921816, 5.14089209357940565142487269204, 6.53626902481560543000541953978, 7.02667120384222301206390105827, 7.81215193802985610959684826683, 8.646255196618321754286567869918, 9.866356913518640852382183339820

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