Properties

Label 2-1950-1.1-c1-0-14
Degree $2$
Conductor $1950$
Sign $1$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s − 4·7-s + 8-s + 9-s + 4·11-s + 12-s + 13-s − 4·14-s + 16-s − 4·17-s + 18-s + 7·19-s − 4·21-s + 4·22-s + 4·23-s + 24-s + 26-s + 27-s − 4·28-s + 5·29-s + 4·31-s + 32-s + 4·33-s − 4·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 0.277·13-s − 1.06·14-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 1.60·19-s − 0.872·21-s + 0.852·22-s + 0.834·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.755·28-s + 0.928·29-s + 0.718·31-s + 0.176·32-s + 0.696·33-s − 0.685·34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1950} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.248449200\)
\(L(\frac12)\) \(\approx\) \(3.248449200\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 9 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 12 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−9.289973885005408013168634093922, −8.518131638991593059128101493380, −7.37533693605609519561095085253, −6.63125696830904895702083084697, −6.25625602915082758497012566096, −5.04165599180722531782139771323, −4.05388339022438793019147473286, −3.31769869380866364367700498552, −2.65392302601634015359034046378, −1.13552684924521539150569858879, 1.13552684924521539150569858879, 2.65392302601634015359034046378, 3.31769869380866364367700498552, 4.05388339022438793019147473286, 5.04165599180722531782139771323, 6.25625602915082758497012566096, 6.63125696830904895702083084697, 7.37533693605609519561095085253, 8.518131638991593059128101493380, 9.289973885005408013168634093922

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