Properties

Label 2-1150-23.22-c2-0-13
Degree $2$
Conductor $1150$
Sign $-0.684 - 0.729i$
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 + 0.278·3-s + 2.00·4-s + 0.393·6-s + 8.51i·7-s + 2.82·8-s − 8.92·9-s + 7.57i·11-s + 0.557·12-s + 2.64·13-s + 12.0i·14-s + 4.00·16-s + 7.56i·17-s − 12.6·18-s − 24.2i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.0928·3-s + 0.500·4-s + 0.0656·6-s + 1.21i·7-s + 0.353·8-s − 0.991·9-s + 0.688i·11-s + 0.0464·12-s + 0.203·13-s + 0.859i·14-s + 0.250·16-s + 0.444i·17-s − 0.701·18-s − 1.27i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.684 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $-0.684 - 0.729i$
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: \((2,\ 1150,\ (\ :1),\ -0.684 - 0.729i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.786888065\)
\(L(\frac12)\) \(\approx\) \(1.786888065\)
\(L(2)\) 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 \( 1 + (-15.7 - 16.7i)T \)
good3 \( 1 - 0.278T + 9T^{2} \)
7 \( 1 - 8.51iT - 49T^{2} \)
11 \( 1 - 7.57iT - 121T^{2} \)
13 \( 1 - 2.64T + 169T^{2} \)
17 \( 1 - 7.56iT - 289T^{2} \)
19 \( 1 + 24.2iT - 361T^{2} \)
29 \( 1 + 31.8T + 841T^{2} \)
31 \( 1 + 56.5T + 961T^{2} \)
37 \( 1 - 39.9iT - 1.36e3T^{2} \)
41 \( 1 + 42.5T + 1.68e3T^{2} \)
43 \( 1 - 20.5iT - 1.84e3T^{2} \)
47 \( 1 + 84.3T + 2.20e3T^{2} \)
53 \( 1 - 11.9iT - 2.80e3T^{2} \)
59 \( 1 - 67.6T + 3.48e3T^{2} \)
61 \( 1 - 35.1iT - 3.72e3T^{2} \)
67 \( 1 - 44.0iT - 4.48e3T^{2} \)
71 \( 1 - 8.86T + 5.04e3T^{2} \)
73 \( 1 - 87.4T + 5.32e3T^{2} \)
79 \( 1 - 154. iT - 6.24e3T^{2} \)
83 \( 1 + 141. iT - 6.88e3T^{2} \)
89 \( 1 - 63.7iT - 7.92e3T^{2} \)
97 \( 1 - 143. iT - 9.40e3T^{2} \)
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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

−9.789120417804372318618898050372, −9.038735501839854147539195009783, −8.368005993726622436922819402042, −7.28866430507277141054385431814, −6.42509206834540944312457235304, −5.45920210866195469095515909583, −5.03652901989488182435859043793, −3.64429561683467514420810655681, −2.75930472588587731238255610831, −1.81612518106814314088874695274, 0.37704120309360696647356818772, 1.90383510043984212989148360696, 3.36784216280523958970170025231, 3.76631475221548685034163006764, 5.07102332081577760162183336927, 5.79425350737820330294284535108, 6.73704535797151084398860468585, 7.56302964392880724526256414066, 8.360392048843908886601228677310, 9.295522350356606034460273978856

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