Properties

Label 2-1950-5.4-c1-0-2
Degree $2$
Conductor $1950$
Sign $0.447 - 0.894i$
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  i·2-s i·3-s − 4-s − 6-s + i·7-s + i·8-s − 9-s − 5·11-s + i·12-s i·13-s + 14-s + 16-s − 5i·17-s + i·18-s + 21-s + 5i·22-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.377i·7-s + 0.353i·8-s − 0.333·9-s − 1.50·11-s + 0.288i·12-s − 0.277i·13-s + 0.267·14-s + 0.250·16-s − 1.21i·17-s + 0.235i·18-s + 0.218·21-s + 1.06i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3982971107\)
\(L(\frac12)\) \(\approx\) \(0.3982971107\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + iT \)
good7 \( 1 - iT - 7T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
17 \( 1 + 5iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 7T + 29T^{2} \)
31 \( 1 + 9T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 9iT - 47T^{2} \)
53 \( 1 - 11iT - 53T^{2} \)
59 \( 1 + T + 59T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 - 15iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 + 16T + 89T^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−9.358836541444472762023956318257, −8.560268597684493583990931931554, −7.81435049446838867369636564597, −7.16495357073504651045490759837, −5.99122045962487592630226658800, −5.25164082612100078630129655200, −4.48536724697619186947534976668, −2.91802923056800729184531683581, −2.68248858013743886723859883368, −1.27164726816683742350119633728, 0.14732165815762724911452578913, 2.07935729728428045054184120617, 3.41260050546260402018528298376, 4.18747228020501973519725178121, 5.16985933931257427817887475425, 5.69019976214886025208218714122, 6.72851077647448023587886233824, 7.51944799593593933739739001118, 8.268475352722482195552890199545, 8.880145600659618493736890585642

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