↑

Properties

Label 2-1700-1700.219-c0-0-0
Degree $2$
Conductor $1700$
Sign $0.321 - 0.946i$
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  + (−0.891 − 0.453i)2-s + (0.587 + 0.809i)4-s + (0.156 + 0.987i)5-s + (−0.156 − 0.987i)8-s + (−0.453 − 0.891i)9-s + (0.309 − 0.951i)10-s + (−0.610 + 1.87i)13-s + (−0.309 + 0.951i)16-s + (0.587 + 0.809i)17-s + 1.00i·18-s + (−0.707 + 0.707i)20-s + (−0.951 + 0.309i)25-s + (1.39 − 1.39i)26-s + (0.678 − 1.10i)29-s + (0.707 − 0.707i)32-s + ⋯
L(s)  = 1  + (−0.891 − 0.453i)2-s + (0.587 + 0.809i)4-s + (0.156 + 0.987i)5-s + (−0.156 − 0.987i)8-s + (−0.453 − 0.891i)9-s + (0.309 − 0.951i)10-s + (−0.610 + 1.87i)13-s + (−0.309 + 0.951i)16-s + (0.587 + 0.809i)17-s + 1.00i·18-s + (−0.707 + 0.707i)20-s + (−0.951 + 0.309i)25-s + (1.39 − 1.39i)26-s + (0.678 − 1.10i)29-s + (0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1700\)    =    \(2^{2} \cdot 5^{2} \cdot 17\)
Sign: $0.321 - 0.946i$
Analytic conductor: \(0.848410\)
Root analytic conductor: \(0.921092\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1700} (219, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1700,\ (\ :0),\ 0.321 - 0.946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6298764161\)
\(L(\frac12)\) \(\approx\) \(0.6298764161\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.891 + 0.453i)T \)
5 \( 1 + (-0.156 - 0.987i)T \)
17 \( 1 + (-0.587 - 0.809i)T \)
good3 \( 1 + (0.453 + 0.891i)T^{2} \)
7 \( 1 + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.987 - 0.156i)T^{2} \)
13 \( 1 + (0.610 - 1.87i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.951 - 0.309i)T^{2} \)
23 \( 1 + (0.987 + 0.156i)T^{2} \)
29 \( 1 + (-0.678 + 1.10i)T + (-0.453 - 0.891i)T^{2} \)
31 \( 1 + (0.891 + 0.453i)T^{2} \)
37 \( 1 + (-0.156 - 1.98i)T + (-0.987 + 0.156i)T^{2} \)
41 \( 1 + (0.678 - 0.794i)T + (-0.156 - 0.987i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.0489 - 0.309i)T + (-0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.587 - 0.809i)T^{2} \)
61 \( 1 + (0.133 - 1.70i)T + (-0.987 - 0.156i)T^{2} \)
67 \( 1 + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (-0.453 - 0.891i)T^{2} \)
73 \( 1 + (0.355 - 0.303i)T + (0.156 - 0.987i)T^{2} \)
79 \( 1 + (0.891 - 0.453i)T^{2} \)
83 \( 1 + (-0.951 + 0.309i)T^{2} \)
89 \( 1 + (-0.863 + 0.280i)T + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.133 - 0.0819i)T + (0.453 + 0.891i)T^{2} \)
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

−9.960986552539537321240042968232, −8.991505647407476249296442685980, −8.261126700119648195222669320440, −7.32409641793615494426424350517, −6.54480319963182285800858624849, −6.16284727622219993066369609558, −4.44717798319235665433057813771, −3.50522680931342568913039156944, −2.65720574737630399167817138641, −1.59517129940446703033594456416, 0.64469698840076051382600093762, 2.06946902812808537977710922995, 3.14344746995284566542877221915, 5.00313887829000998995451102994, 5.22865929413271943713281462763, 6.06803249364087724754721463771, 7.41475349296843563016870800035, 7.80422607845232957776284886377, 8.554859909884754417807897876082, 9.233747905621321413212207371695

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