Properties

Label 2-5850-1.1-c1-0-13
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 − 4·7-s + 8-s − 2·11-s − 13-s − 4·14-s + 16-s + 4·17-s + 2·19-s − 2·22-s − 6·23-s − 26-s − 4·28-s + 2·29-s − 4·31-s + 32-s + 4·34-s + 6·37-s + 2·38-s + 6·41-s − 8·43-s − 2·44-s − 6·46-s + 8·47-s + 9·49-s − 52-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s − 0.603·11-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 0.970·17-s + 0.458·19-s − 0.426·22-s − 1.25·23-s − 0.196·26-s − 0.755·28-s + 0.371·29-s − 0.718·31-s + 0.176·32-s + 0.685·34-s + 0.986·37-s + 0.324·38-s + 0.937·41-s − 1.21·43-s − 0.301·44-s − 0.884·46-s + 1.16·47-s + 9/7·49-s − 0.138·52-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

\(L(1)\) \(\approx\) \(2.248658230\)
\(L(\frac12)\) \(\approx\) \(2.248658230\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−7.87498419555034890742165610765, −7.29485373851164431988882778158, −6.60290116620330028250116272132, −5.77144927324143348913963681994, −5.48331297526468208290874126097, −4.32578104712105166619371068892, −3.62643806926958717220404425149, −2.94308915875784264076705004238, −2.18984263408584192767406707815, −0.68431411467238404496665432852, 0.68431411467238404496665432852, 2.18984263408584192767406707815, 2.94308915875784264076705004238, 3.62643806926958717220404425149, 4.32578104712105166619371068892, 5.48331297526468208290874126097, 5.77144927324143348913963681994, 6.60290116620330028250116272132, 7.29485373851164431988882778158, 7.87498419555034890742165610765

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