Properties

Label 2-1950-1.1-c1-0-36
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 $1$

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 − 6·11-s + 12-s + 13-s − 4·14-s + 16-s − 4·17-s + 18-s + 2·19-s − 4·21-s − 6·22-s − 6·23-s + 24-s + 26-s + 27-s − 4·28-s − 10·29-s + 4·31-s + 32-s − 6·33-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 − 1.80·11-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 + 0.458·19-s − 0.872·21-s − 1.27·22-s − 1.25·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.755·28-s − 1.85·29-s + 0.718·31-s + 0.176·32-s − 1.04·33-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: \(1\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.872719737543440996534603832810, −7.76289295114252606314765360714, −7.32363828755953441216992328393, −6.22154667625865130162104138279, −5.70958952817226870085238408234, −4.59257511667402952390000494837, −3.65328685323855751887250626877, −2.90324884014687840380390019235, −2.13537777044490151588674952443, 0, 2.13537777044490151588674952443, 2.90324884014687840380390019235, 3.65328685323855751887250626877, 4.59257511667402952390000494837, 5.70958952817226870085238408234, 6.22154667625865130162104138279, 7.32363828755953441216992328393, 7.76289295114252606314765360714, 8.872719737543440996534603832810

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