↑

Properties

Label 2-444-111.44-c0-0-0
Degree $2$
Conductor $444$
Sign $0.957 - 0.287i$
Analytic cond. $0.221584$
Root an. cond. $0.470728$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 − 0.342i)3-s + (0.266 + 1.50i)7-s + (0.766 + 0.642i)9-s + (1.17 − 0.984i)13-s + (0.939 + 0.342i)19-s + (0.266 − 1.50i)21-s + (−0.939 + 0.342i)25-s + (−0.500 − 0.866i)27-s + 0.347·31-s + (−0.5 + 0.866i)37-s + (−1.43 + 0.524i)39-s − 1.87·43-s + (−1.26 + 0.460i)49-s + (−0.766 − 0.642i)57-s + (1.53 − 1.28i)61-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)3-s + (0.266 + 1.50i)7-s + (0.766 + 0.642i)9-s + (1.17 − 0.984i)13-s + (0.939 + 0.342i)19-s + (0.266 − 1.50i)21-s + (−0.939 + 0.342i)25-s + (−0.500 − 0.866i)27-s + 0.347·31-s + (−0.5 + 0.866i)37-s + (−1.43 + 0.524i)39-s − 1.87·43-s + (−1.26 + 0.460i)49-s + (−0.766 − 0.642i)57-s + (1.53 − 1.28i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $0.957 - 0.287i$
Analytic conductor: \(0.221584\)
Root analytic conductor: \(0.470728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 444,\ (\ :0),\ 0.957 - 0.287i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6958094810\)
\(L(\frac12)\) \(\approx\) \(0.6958094810\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.939 + 0.342i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (0.939 - 0.342i)T^{2} \)
7 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
17 \( 1 + (-0.173 - 0.984i)T^{2} \)
19 \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - 0.347T + T^{2} \)
41 \( 1 + (-0.173 + 0.984i)T^{2} \)
43 \( 1 + 1.87T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T^{2} \)
59 \( 1 + (0.939 + 0.342i)T^{2} \)
61 \( 1 + (-1.53 + 1.28i)T + (0.173 - 0.984i)T^{2} \)
67 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + 1.87T + T^{2} \)
79 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.173 - 0.984i)T^{2} \)
89 \( 1 + (0.939 + 0.342i)T^{2} \)
97 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.67091039348969310434297221744, −10.61161937704137275876292388964, −9.685310171327389148291418844719, −8.514544371122196309628831432002, −7.79779190284520659171390217628, −6.43513087259533752621690969727, −5.69467496485834838357228796758, −5.02248410275126686901703221309, −3.30858617282751880688394370469, −1.69417690018706586725595872319, 1.28921091157841504566497699367, 3.72346803676097983848360538601, 4.38136277367718083949512563703, 5.58035420075195208393774082325, 6.68982983227876717679169190660, 7.33683491217547496538576664975, 8.623715007642023812506553696768, 9.820191808826047708040435734133, 10.41864740427759308172055716424, 11.38939209607595444700485532848

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