Properties

Label 2-1950-1.1-c1-0-10
Degree $2$
Conductor $1950$
Sign $1$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
Motivic weight $1$
Arithmetic yes
Rational no
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 + 2.82·7-s − 8-s + 9-s + 5.65·11-s − 12-s + 13-s − 2.82·14-s + 16-s − 0.828·17-s − 18-s + 2.82·19-s − 2.82·21-s − 5.65·22-s + 8.48·23-s + 24-s − 26-s − 27-s + 2.82·28-s − 8.82·29-s + 4·31-s − 32-s − 5.65·33-s + 0.828·34-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s + 1.06·7-s − 0.353·8-s + 0.333·9-s + 1.70·11-s − 0.288·12-s + 0.277·13-s − 0.755·14-s + 0.250·16-s − 0.200·17-s − 0.235·18-s + 0.648·19-s − 0.617·21-s − 1.20·22-s + 1.76·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.534·28-s − 1.63·29-s + 0.718·31-s − 0.176·32-s − 0.984·33-s + 0.142·34-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: no
Arithmetic: yes
Character: $\chi_{1950} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.407856191\)
\(L(\frac12)\) \(\approx\) \(1.407856191\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.82T + 7T^{2} \)
11 \( 1 - 5.65T + 11T^{2} \)
17 \( 1 + 0.828T + 17T^{2} \)
19 \( 1 - 2.82T + 19T^{2} \)
23 \( 1 - 8.48T + 23T^{2} \)
29 \( 1 + 8.82T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 - 11.6T + 37T^{2} \)
41 \( 1 + 7.65T + 41T^{2} \)
43 \( 1 + 9.65T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 13.3T + 53T^{2} \)
59 \( 1 + 2.34T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 5.65T + 67T^{2} \)
71 \( 1 + 5.65T + 71T^{2} \)
73 \( 1 - 14.4T + 73T^{2} \)
79 \( 1 - 2.34T + 79T^{2} \)
83 \( 1 + 6.34T + 83T^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 - 3.17T + 97T^{2} \)
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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

−9.219115903038074626288763489422, −8.497802958648184043353661337669, −7.61867841167693829596398597289, −6.88180835921894957660755905301, −6.18436617040436159419623089311, −5.19307669183466344828246987613, −4.36491764852097863929258806300, −3.27662068984700935406229227369, −1.73887831165524401257091125502, −1.01601587301387497534768664382, 1.01601587301387497534768664382, 1.73887831165524401257091125502, 3.27662068984700935406229227369, 4.36491764852097863929258806300, 5.19307669183466344828246987613, 6.18436617040436159419623089311, 6.88180835921894957660755905301, 7.61867841167693829596398597289, 8.497802958648184043353661337669, 9.219115903038074626288763489422

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