Properties

Label 2-1150-115.114-c2-0-70
Degree $2$
Conductor $1150$
Sign $-0.659 - 0.751i$
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  + 1.41i·2-s − 5.41i·3-s − 2.00·4-s + 7.66·6-s − 8.24·7-s − 2.82i·8-s − 20.3·9-s − 15.8i·11-s + 10.8i·12-s − 14.3i·13-s − 11.6i·14-s + 4.00·16-s + 10.1·17-s − 28.8i·18-s − 36.5i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.80i·3-s − 0.500·4-s + 1.27·6-s − 1.17·7-s − 0.353i·8-s − 2.26·9-s − 1.43i·11-s + 0.903i·12-s − 1.10i·13-s − 0.832i·14-s + 0.250·16-s + 0.598·17-s − 1.60i·18-s − 1.92i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $-0.659 - 0.751i$
Analytic conductor: \(31.3352\)
Root analytic conductor: \(5.59778\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1150} (1149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1150,\ (\ :1),\ -0.659 - 0.751i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5734869102\)
\(L(\frac12)\) \(\approx\) \(0.5734869102\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
5 \( 1 \)
23 \( 1 + (-5.84 - 22.2i)T \)
good3 \( 1 + 5.41iT - 9T^{2} \)
7 \( 1 + 8.24T + 49T^{2} \)
11 \( 1 + 15.8iT - 121T^{2} \)
13 \( 1 + 14.3iT - 169T^{2} \)
17 \( 1 - 10.1T + 289T^{2} \)
19 \( 1 + 36.5iT - 361T^{2} \)
29 \( 1 + 6.46T + 841T^{2} \)
31 \( 1 + 42.8T + 961T^{2} \)
37 \( 1 - 63.6T + 1.36e3T^{2} \)
41 \( 1 + 37.0T + 1.68e3T^{2} \)
43 \( 1 - 6.00T + 1.84e3T^{2} \)
47 \( 1 - 32.4iT - 2.20e3T^{2} \)
53 \( 1 + 36.6T + 2.80e3T^{2} \)
59 \( 1 + 6.65T + 3.48e3T^{2} \)
61 \( 1 - 55.7iT - 3.72e3T^{2} \)
67 \( 1 + 4.45T + 4.48e3T^{2} \)
71 \( 1 - 118.T + 5.04e3T^{2} \)
73 \( 1 - 82.2iT - 5.32e3T^{2} \)
79 \( 1 + 133. iT - 6.24e3T^{2} \)
83 \( 1 + 67.5T + 6.88e3T^{2} \)
89 \( 1 - 104. iT - 7.92e3T^{2} \)
97 \( 1 - 98.6T + 9.40e3T^{2} \)
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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

−8.846717595106741328053315140710, −7.983470980822108047681728967180, −7.37652424494484555043846912388, −6.62149897073662169515831037080, −5.94104539244640185777737044875, −5.35812463680895205849546020618, −3.37469298805678506035394404536, −2.77022342487484879397291434587, −0.975089854818079972994523632449, −0.20804400515207799442388419970, 2.04654846066844308254901950666, 3.31031582030877198039608464512, 3.93868110369208437828771415989, 4.64374455271030809823886187777, 5.61976873697064575707121030922, 6.61111380028720674267159715401, 7.941870973435789671788052567333, 9.052128913444928234495734580269, 9.652315819364529933419949087324, 9.941859226492720282585192571576

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