Properties

Label 2-1800-8.5-c1-0-77
Degree $2$
Conductor $1800$
Sign $i$
Analytic cond. $14.3730$
Root an. cond. $3.79118$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 0.707i)2-s + (0.999 − 1.73i)4-s + 2.44·7-s − 2.82i·8-s − 3.46i·11-s + (2.99 − 1.73i)14-s + (−2.00 − 3.46i)16-s + 4.89·17-s + 3.46i·19-s + (−2.44 − 4.24i)22-s − 2.44·23-s + (2.44 − 4.24i)28-s + 4·31-s + (−4.89 − 2.82i)32-s + (5.99 − 3.46i)34-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)2-s + (0.499 − 0.866i)4-s + 0.925·7-s − 0.999i·8-s − 1.04i·11-s + (0.801 − 0.462i)14-s + (−0.500 − 0.866i)16-s + 1.18·17-s + 0.794i·19-s + (−0.522 − 0.904i)22-s − 0.510·23-s + (0.462 − 0.801i)28-s + 0.718·31-s + (−0.866 − 0.499i)32-s + (1.02 − 0.594i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.284142281\)
\(L(\frac12)\) \(\approx\) \(3.284142281\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 2.44T + 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 4.89T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 2.44T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 8.48iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 4.24iT - 43T^{2} \)
47 \( 1 + 7.34T + 47T^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 + 3.46iT - 61T^{2} \)
67 \( 1 + 4.24iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 4.89T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 9.89iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 4.89T + 97T^{2} \)
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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

−9.187807309129277302559863130424, −8.163597639970911579334014266459, −7.57362387476304184414329317022, −6.36381441060053293493386414584, −5.68424919781061868456256601689, −5.01399850580831799103531309427, −3.97227756895678813657709949533, −3.24751893067686940059930326687, −2.06132044457140377327154492673, −0.983611079417861409779447656805, 1.60717016134406587398928268501, 2.70613910550544931324325307334, 3.78868240524736559286547624578, 4.82843749293151684046092018481, 5.11235437880107880474462453313, 6.31099379856755698196844356337, 6.98476350071855637160169008248, 7.967175532517274773062957161707, 8.193699899384178441402860914879, 9.480093005510564810500590099112

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