↑

Properties

Label 2-1700-340.339-c0-0-3
Degree $2$
Conductor $1700$
Sign $0.894 + 0.447i$
Analytic cond. $0.848410$
Root an. cond. $0.921092$
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·2-s − 4-s + i·8-s + 9-s + 2i·13-s + 16-s + i·17-s − i·18-s + 2·26-s − i·32-s + 34-s − 36-s + 49-s − 2i·52-s − 2i·53-s + ⋯
L(s)  = 1  − i·2-s − 4-s + i·8-s + 9-s + 2i·13-s + 16-s + i·17-s − i·18-s + 2·26-s − i·32-s + 34-s − 36-s + 49-s − 2i·52-s − 2i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(0.848410\)
Root analytic conductor: \(0.921092\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (1699, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :0),\ 0.894 + 0.447i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.058829339\)
\(L(\frac12)\) \(\approx\) \(1.058829339\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
17 \( 1 - iT \)
good3 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 2iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 2iT - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 + T^{2} \)
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

−9.550987724788604379431264379495, −8.947034852315499554544232547344, −8.109091838592266158893116795767, −7.07884851431177866454503796137, −6.29059528335442226075537437258, −5.06841425462184414006062935921, −4.21931866556954222613572583014, −3.70773392692290291612075294082, −2.22085192687463278883090077162, −1.48014408051926811139872589975, 0.948393339903121943391961748045, 2.84927817052814625843250387070, 3.90192539579700297486776828678, 4.87696887015863872538826880717, 5.53969064161163281918024913047, 6.41533775595560043029732049518, 7.43011918093119782203783332410, 7.69605804913797349617944069976, 8.685197224461549113258892412836, 9.493249164785248564425325113513

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