Properties

Label 2-1950-65.64-c1-0-27
Degree $2$
Conductor $1950$
Sign $0.813 - 0.581i$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
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  + 2-s + i·3-s + 4-s + i·6-s + 5.12·7-s + 8-s − 9-s + 3.12i·11-s + i·12-s + (3.56 − 0.561i)13-s + 5.12·14-s + 16-s − 2i·17-s − 18-s − 6i·19-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577i·3-s + 0.5·4-s + 0.408i·6-s + 1.93·7-s + 0.353·8-s − 0.333·9-s + 0.941i·11-s + 0.288i·12-s + (0.987 − 0.155i)13-s + 1.36·14-s + 0.250·16-s − 0.485i·17-s − 0.235·18-s − 1.37i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.813 - 0.581i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.813 - 0.581i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.573644048\)
\(L(\frac12)\) \(\approx\) \(3.573644048\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - iT \)
5 \( 1 \)
13 \( 1 + (-3.56 + 0.561i)T \)
good7 \( 1 - 5.12T + 7T^{2} \)
11 \( 1 - 3.12iT - 11T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 - 3.12iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 5.12iT - 31T^{2} \)
37 \( 1 + 3.12T + 37T^{2} \)
41 \( 1 + 9.12iT - 41T^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 + 10.2T + 47T^{2} \)
53 \( 1 - 11.3iT - 53T^{2} \)
59 \( 1 - 7.12iT - 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + 13.1T + 67T^{2} \)
71 \( 1 - 6.24iT - 71T^{2} \)
73 \( 1 + 4.87T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + 5.12iT - 89T^{2} \)
97 \( 1 - 4.87T + 97T^{2} \)
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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

−9.046532923520976304624704439827, −8.608910184356256243031300992436, −7.52561886941854395807025956389, −7.04145283102416146997108320783, −5.69920461127979733399294528878, −5.05152443058114965860914500734, −4.52733588851498654606123115402, −3.65210333893476663583269135057, −2.39414728000065705917862966864, −1.40705391671701071882904796551, 1.28432688322661936306282395102, 1.94493432342081238329484843558, 3.31142390486135480103896126337, 4.22015431796780892732605626769, 5.09941659616532538390615043899, 5.93324231701598114204043984664, 6.48127925272364125998217491812, 7.81700528020548538854474676703, 8.092580995260058328677590795611, 8.702855377378708212296254382845

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