↑

Properties

Label 2-1800-15.14-c0-0-2
Degree $2$
Conductor $1800$
Sign $0.881 + 0.472i$
Analytic cond. $0.898317$
Root an. cond. $0.947795$
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  − i·7-s + 1.41i·11-s − i·13-s + 1.41·17-s − 19-s + 1.41·23-s − 1.41i·29-s + 31-s − i·43-s − 1.41·47-s + 1.41i·59-s + 61-s − i·67-s + 1.41·77-s − 1.41·83-s + ⋯
L(s)  = 1  − i·7-s + 1.41i·11-s − i·13-s + 1.41·17-s − 19-s + 1.41·23-s − 1.41i·29-s + 31-s − i·43-s − 1.41·47-s + 1.41i·59-s + 61-s − i·67-s + 1.41·77-s − 1.41·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.881 + 0.472i$
Analytic conductor: \(0.898317\)
Root analytic conductor: \(0.947795\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :0),\ 0.881 + 0.472i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.171917278\)
\(L(\frac12)\) \(\approx\) \(1.171917278\)
\(L(1)\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 + iT - T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 - 1.41T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 1.41iT - T^{2} \)
61 \( 1 - T + T^{2} \)
67 \( 1 + iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - iT - T^{2} \)
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

−9.652898148737461879963248236734, −8.495122324399266807889997268020, −7.70941696810620738849954989921, −7.18415047070095548953211560849, −6.30258738513444534841687974791, −5.23004398713219354713466883554, −4.46968798262192134968937288485, −3.57557658521357573758159374877, −2.46052255873522522106509496076, −1.05848232644728605482463645322, 1.37440935904050908754920087612, 2.75623640170741714678183942917, 3.46313536411105227985050275299, 4.73731543721995202743217354000, 5.53844628499141853207671980768, 6.28292280200242889734786068729, 7.04093209317395606947563179057, 8.281688351888695183429605264067, 8.625066010674085733002667837817, 9.396743263738208099911690800017

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