Properties

Label 2-5850-5.4-c1-0-18
Degree $2$
Conductor $5850$
Sign $-0.447 - 0.894i$
Analytic cond. $46.7124$
Root an. cond. $6.83465$
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  + i·2-s − 4-s i·8-s − 4·11-s + i·13-s + 16-s − 6i·17-s − 4·19-s − 4i·22-s − 8i·23-s − 26-s + 6·29-s − 8·31-s + i·32-s + 6·34-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.353i·8-s − 1.20·11-s + 0.277i·13-s + 0.250·16-s − 1.45i·17-s − 0.917·19-s − 0.852i·22-s − 1.66i·23-s − 0.196·26-s + 1.11·29-s − 1.43·31-s + 0.176i·32-s + 1.02·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5850\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 13\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(46.7124\)
Root analytic conductor: \(6.83465\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5850} (5149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5850,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.012050612\)
\(L(\frac12)\) \(\approx\) \(1.012050612\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 - iT \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 16T + 71T^{2} \)
73 \( 1 - 2iT - 73T^{2} \)
79 \( 1 - 16T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−8.286258709119107122630602315765, −7.56849210619634473157221399917, −6.93914798720800732800545544368, −6.28324191211486244927667813695, −5.48428650712025179587996566552, −4.73733487755462747022174169327, −4.27295905786050973221422936989, −2.94880541408032256572315728980, −2.38714043190184018090325585699, −0.814609396860924717355402228032, 0.33460377106880606640827181061, 1.73578775414940570805044367105, 2.36343578402863494635925157362, 3.45101078527810027276363554517, 3.96175539939132962797576294529, 4.98373900124573529542895096691, 5.62036225757663055686889139876, 6.26791664491462523910704401979, 7.48152170200871643282777105370, 7.81334780575291103524175800491

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