Properties

Label 2-2850-5.4-c1-0-21
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 + 2i·7-s i·8-s − 9-s + 6·11-s + i·12-s + 4i·13-s − 2·14-s + 16-s − 6i·17-s i·18-s − 19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.755i·7-s − 0.353i·8-s − 0.333·9-s + 1.80·11-s + 0.288i·12-s + 1.10i·13-s − 0.534·14-s + 0.250·16-s − 1.45i·17-s − 0.235i·18-s − 0.229·19-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

\(L(1)\) \(\approx\) \(1.850663933\)
\(L(\frac12)\) \(\approx\) \(1.850663933\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 7T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 16iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−8.838180755625629281954261427003, −8.245815574838388641102914317399, −7.12005626896697026260354829827, −6.67944941443376805957723470291, −6.17319985475513657134332534072, −5.10832060079206385189879465539, −4.38784166328517770241147396801, −3.33221020521897279673932139690, −2.15785556659230915868782397180, −1.03701669191881756260791499578, 0.75488886847315970442282072106, 1.81211727436000610200546691753, 3.18121582658307778536870310425, 3.89444916056887763846471219301, 4.34385956054244529738492942770, 5.47148481491704216726392496216, 6.29231337625413764445201314182, 7.10038595559102844680560837326, 8.279821640476004608263266026263, 8.633953515486354492096208844947

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