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Properties

Label 2-2850-5.4-c1-0-7
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 + 2·11-s − i·12-s + 4i·13-s + 2·14-s + 16-s + 2i·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 + 0.603·11-s − 0.288i·12-s + 1.10i·13-s + 0.534·14-s + 0.250·16-s + 0.485i·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

\(L(1)\) \(\approx\) \(1.038802969\)
\(L(\frac12)\) \(\approx\) \(1.038802969\)
\(L(\frac{3}{2})\) 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 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + 6iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−9.156799685391133279097436904425, −8.680154042014038424836184127659, −7.63566712124265102626018713438, −6.68195493960369995816912778209, −5.73851389389688995775799905267, −5.10773717995314291651008490744, −4.04496712665550083954375539417, −3.57744448825122072318256757528, −2.37937152391413829639298038725, −1.53386947559893321155787256698, 0.33966234015730159263296938501, 1.44941456456430841887347479534, 2.91850913226440531604287501939, 3.79104109000987595875768053949, 4.79431436958230531772701946549, 5.56601505946024522728239876014, 6.42002802373597097689393308297, 7.07669574052699122284242608032, 7.62667610961840292057413200458, 8.404032806921054579354080136976

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