Properties

Label 2-235950-1.1-c1-0-97
Degree $2$
Conductor $235950$
Sign $1$
Analytic cond. $1884.07$
Root an. cond. $43.4058$
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 + 2·7-s + 8-s + 9-s + 12-s − 13-s + 2·14-s + 16-s + 8·17-s + 18-s + 6·19-s + 2·21-s − 6·23-s + 24-s − 26-s + 27-s + 2·28-s + 4·29-s + 32-s + 8·34-s + 36-s + 2·37-s + 6·38-s − 39-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s − 0.277·13-s + 0.534·14-s + 1/4·16-s + 1.94·17-s + 0.235·18-s + 1.37·19-s + 0.436·21-s − 1.25·23-s + 0.204·24-s − 0.196·26-s + 0.192·27-s + 0.377·28-s + 0.742·29-s + 0.176·32-s + 1.37·34-s + 1/6·36-s + 0.328·37-s + 0.973·38-s − 0.160·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 235950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(235950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 11^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(1884.07\)
Root analytic conductor: \(43.4058\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{235950} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 235950,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.545483514\)
\(L(\frac12)\) \(\approx\) \(8.545483514\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 \)
13 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
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

−13.05328914498867, −12.34812490329580, −11.93610791562620, −11.74135484813272, −11.30362350541446, −10.34472383824925, −10.18594292892573, −9.849137673268797, −9.148570207429703, −8.594429552555829, −7.999024089004384, −7.748081281913722, −7.325602917534633, −6.812546115143960, −5.950564267730077, −5.718259758581788, −5.128938395249999, −4.686299346676414, −4.078204518575932, −3.542676600158720, −3.077455181060956, −2.567593995118365, −1.838168178297979, −1.317497144531320, −0.7061654932047908, 0.7061654932047908, 1.317497144531320, 1.838168178297979, 2.567593995118365, 3.077455181060956, 3.542676600158720, 4.078204518575932, 4.686299346676414, 5.128938395249999, 5.718259758581788, 5.950564267730077, 6.812546115143960, 7.325602917534633, 7.748081281913722, 7.999024089004384, 8.594429552555829, 9.148570207429703, 9.849137673268797, 10.18594292892573, 10.34472383824925, 11.30362350541446, 11.74135484813272, 11.93610791562620, 12.34812490329580, 13.05328914498867

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