↑

Properties

Label 2-1200-5.2-c2-0-17
Degree $2$
Conductor $1200$
Sign $0.973 - 0.229i$
Analytic cond. $32.6976$
Root an. cond. $5.71818$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 − 1.22i)3-s + (7.34 + 7.34i)7-s − 2.99i·9-s + 18·11-s + (7.34 − 7.34i)13-s + (4.89 + 4.89i)17-s + 10i·19-s + 18·21-s + (−19.5 + 19.5i)23-s + (−3.67 − 3.67i)27-s − 22·31-s + (22.0 − 22.0i)33-s + (−7.34 − 7.34i)37-s − 18i·39-s − 18·41-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (1.04 + 1.04i)7-s − 0.333i·9-s + 1.63·11-s + (0.565 − 0.565i)13-s + (0.288 + 0.288i)17-s + 0.526i·19-s + 0.857·21-s + (−0.851 + 0.851i)23-s + (−0.136 − 0.136i)27-s − 0.709·31-s + (0.668 − 0.668i)33-s + (−0.198 − 0.198i)37-s − 0.461i·39-s − 0.439·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(32.6976\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :1),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.943018283\)
\(L(\frac12)\) \(\approx\) \(2.943018283\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.22 + 1.22i)T \)
5 \( 1 \)
good7 \( 1 + (-7.34 - 7.34i)T + 49iT^{2} \)
11 \( 1 - 18T + 121T^{2} \)
13 \( 1 + (-7.34 + 7.34i)T - 169iT^{2} \)
17 \( 1 + (-4.89 - 4.89i)T + 289iT^{2} \)
19 \( 1 - 10iT - 361T^{2} \)
23 \( 1 + (19.5 - 19.5i)T - 529iT^{2} \)
29 \( 1 - 841T^{2} \)
31 \( 1 + 22T + 961T^{2} \)
37 \( 1 + (7.34 + 7.34i)T + 1.36e3iT^{2} \)
41 \( 1 + 18T + 1.68e3T^{2} \)
43 \( 1 + (-29.3 + 29.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (-44.0 - 44.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (4.89 - 4.89i)T - 2.80e3iT^{2} \)
59 \( 1 - 90iT - 3.48e3T^{2} \)
61 \( 1 - 2T + 3.72e3T^{2} \)
67 \( 1 + (-44.0 - 44.0i)T + 4.48e3iT^{2} \)
71 \( 1 + 72T + 5.04e3T^{2} \)
73 \( 1 + (-44.0 + 44.0i)T - 5.32e3iT^{2} \)
79 \( 1 + 70iT - 6.24e3T^{2} \)
83 \( 1 + (-53.8 + 53.8i)T - 6.88e3iT^{2} \)
89 \( 1 + 90iT - 7.92e3T^{2} \)
97 \( 1 + (-102. - 102. i)T + 9.40e3iT^{2} \)
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

−9.257392035180423329545147059284, −8.799512134677232990469877838884, −8.032685124503597560666448917620, −7.27438180797180889551118520195, −6.06084384357623755753635283428, −5.61670696789413310450753388140, −4.25614257143678954781283947515, −3.39638464842470686883083864571, −2.02453965706723592452500421150, −1.28014291094949043303090298669, 0.987839078480653736463919096631, 2.02672502410707087109300751169, 3.66029235618256847186198273711, 4.16136351275152470838596548232, 5.00982737897083204397340122197, 6.33981180793681909762121562040, 7.07520676344317416378046296162, 7.984436620571488931870607760241, 8.756300033199464188065838719148, 9.443156363310818031778006992349

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