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Properties

Label 2-1150-5.4-c3-0-36
Degree $2$
Conductor $1150$
Sign $0.447 + 0.894i$
Analytic cond. $67.8521$
Root an. cond. $8.23724$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s − 5.04i·3-s − 4·4-s − 10.0·6-s − 5.03i·7-s + 8i·8-s + 1.58·9-s − 5.58·11-s + 20.1i·12-s + 62.7i·13-s − 10.0·14-s + 16·16-s + 19.7i·17-s − 3.17i·18-s − 158.·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.970i·3-s − 0.5·4-s − 0.685·6-s − 0.271i·7-s + 0.353i·8-s + 0.0588·9-s − 0.153·11-s + 0.485i·12-s + 1.33i·13-s − 0.192·14-s + 0.250·16-s + 0.281i·17-s − 0.0416i·18-s − 1.91·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(67.8521\)
Root analytic conductor: \(8.23724\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1150} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1150,\ (\ :3/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.745360913\)
\(L(\frac12)\) \(\approx\) \(1.745360913\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
5 \( 1 \)
23 \( 1 + 23iT \)
good3 \( 1 + 5.04iT - 27T^{2} \)
7 \( 1 + 5.03iT - 343T^{2} \)
11 \( 1 + 5.58T + 1.33e3T^{2} \)
13 \( 1 - 62.7iT - 2.19e3T^{2} \)
17 \( 1 - 19.7iT - 4.91e3T^{2} \)
19 \( 1 + 158.T + 6.85e3T^{2} \)
29 \( 1 - 35.5T + 2.43e4T^{2} \)
31 \( 1 - 282.T + 2.97e4T^{2} \)
37 \( 1 - 139. iT - 5.06e4T^{2} \)
41 \( 1 - 227.T + 6.89e4T^{2} \)
43 \( 1 - 436. iT - 7.95e4T^{2} \)
47 \( 1 + 90.2iT - 1.03e5T^{2} \)
53 \( 1 - 330. iT - 1.48e5T^{2} \)
59 \( 1 - 796.T + 2.05e5T^{2} \)
61 \( 1 + 568.T + 2.26e5T^{2} \)
67 \( 1 + 85.1iT - 3.00e5T^{2} \)
71 \( 1 + 369.T + 3.57e5T^{2} \)
73 \( 1 + 310. iT - 3.89e5T^{2} \)
79 \( 1 - 1.32e3T + 4.93e5T^{2} \)
83 \( 1 + 158. iT - 5.71e5T^{2} \)
89 \( 1 - 1.23e3T + 7.04e5T^{2} \)
97 \( 1 - 106. iT - 9.12e5T^{2} \)
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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

−9.290305176328450450562048531623, −8.445157629495988417023764699533, −7.73082384823926420360272657662, −6.63131016244228753776142352463, −6.24841878112926095282061470450, −4.61058316557143288444632561395, −4.11648213665812794801499454667, −2.60689671665312358013737664324, −1.82830380867609371353478329579, −0.819325275377825849433317100462, 0.57611388679081061203647963712, 2.43349055533756103538051719865, 3.66307200059806434440105768441, 4.48511405111777581556994312739, 5.28703036751915787035029507582, 6.09564101553183982839185087836, 7.06184536201642773396412556968, 8.057104124941433409641839558454, 8.693768301426937709958809224597, 9.517392103449158155262467845472

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