Properties

Label 2-2850-5.4-c1-0-23
Degree $2$
Conductor $2850$
Sign $0.447 - 0.894i$
Analytic cond. $22.7573$
Root an. cond. $4.77046$
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 + i·3-s − 4-s − 6-s i·8-s − 9-s − 4·11-s i·12-s − 2i·13-s + 16-s − 6i·17-s i·18-s + 19-s − 4i·22-s + 4i·23-s + 24-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s − 1.20·11-s − 0.288i·12-s − 0.554i·13-s + 0.250·16-s − 1.45i·17-s − 0.235i·18-s + 0.229·19-s − 0.852i·22-s + 0.834i·23-s + 0.204·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2850\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 19\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(22.7573\)
Root analytic conductor: \(4.77046\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2850} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2850,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.424209793\)
\(L(\frac12)\) \(\approx\) \(1.424209793\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 \)
19 \( 1 - T \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−8.920988396000801812360307286564, −7.965566884637014119472543713347, −7.59214014436594803960150618313, −6.65335616001666438648455720278, −5.67278318263461060108272592286, −5.13618679792012335541814618000, −4.47345439038734654798910190599, −3.29343566030573104969788660387, −2.57104854456918517039530813935, −0.69195047505261628641561516251, 0.76155585672793965265685803370, 2.03633990584666325228816963803, 2.66436393575797093320308151119, 3.79894750409145263244343700128, 4.61499983890732478160884711074, 5.60158468053906102238467638373, 6.27716716059072206147969104682, 7.26062068944332663136375686407, 8.055573476983164953450807104501, 8.552401287104038513288961730484

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