Properties

Label 2-5850-1.1-c1-0-18
Degree $2$
Conductor $5850$
Sign $1$
Analytic cond. $46.7124$
Root an. cond. $6.83465$
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 + 4-s − 2·7-s + 8-s − 4·11-s + 13-s − 2·14-s + 16-s + 8·17-s − 6·19-s − 4·22-s + 6·23-s + 26-s − 2·28-s + 4·29-s + 32-s + 8·34-s + 2·37-s − 6·38-s + 2·41-s + 4·43-s − 4·44-s + 6·46-s − 3·49-s + 52-s − 10·53-s − 2·56-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.755·7-s + 0.353·8-s − 1.20·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 1.94·17-s − 1.37·19-s − 0.852·22-s + 1.25·23-s + 0.196·26-s − 0.377·28-s + 0.742·29-s + 0.176·32-s + 1.37·34-s + 0.328·37-s − 0.973·38-s + 0.312·41-s + 0.609·43-s − 0.603·44-s + 0.884·46-s − 3/7·49-s + 0.138·52-s − 1.37·53-s − 0.267·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5850\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(46.7124\)
Root analytic conductor: \(6.83465\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{5850} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 5850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.727225855\)
\(L(\frac12)\) \(\approx\) \(2.727225855\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
13 \( 1 - T \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−7.79003551516086049978358436531, −7.53237715529879531138905824169, −6.32236866712103413082788478515, −6.12911584329025029627562112611, −5.10344794395496418386832407303, −4.63265733017116256296221359410, −3.40525780893678541212565970767, −3.09880228779188281769753009138, −2.09663474758288346911639767662, −0.77294040377296382231627851684, 0.77294040377296382231627851684, 2.09663474758288346911639767662, 3.09880228779188281769753009138, 3.40525780893678541212565970767, 4.63265733017116256296221359410, 5.10344794395496418386832407303, 6.12911584329025029627562112611, 6.32236866712103413082788478515, 7.53237715529879531138905824169, 7.79003551516086049978358436531

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