↑

Properties

Label 2-840-105.104-c1-0-36
Degree $2$
Conductor $840$
Sign $-0.383 + 0.923i$
Analytic cond. $6.70743$
Root an. cond. $2.58987$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.726 − 1.57i)3-s + (2.20 + 0.341i)5-s + (−2.06 − 1.64i)7-s + (−1.94 + 2.28i)9-s + 1.06i·11-s + 4.82·13-s + (−1.06 − 3.72i)15-s − 7.89i·17-s − 4.02i·19-s + (−1.09 + 4.45i)21-s − 5.69·23-s + (4.76 + 1.50i)25-s + (5.00 + 1.40i)27-s − 2.00i·29-s − 4.89i·31-s + ⋯
L(s)  = 1  + (−0.419 − 0.907i)3-s + (0.988 + 0.152i)5-s + (−0.781 − 0.623i)7-s + (−0.648 + 0.761i)9-s + 0.320i·11-s + 1.33·13-s + (−0.275 − 0.961i)15-s − 1.91i·17-s − 0.922i·19-s + (−0.238 + 0.971i)21-s − 1.18·23-s + (0.953 + 0.301i)25-s + (0.962 + 0.269i)27-s − 0.372i·29-s − 0.879i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.383 + 0.923i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.383 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.721194 - 1.08067i\)
\(L(\frac12)\) \(\approx\) \(0.721194 - 1.08067i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.726 + 1.57i)T \)
5 \( 1 + (-2.20 - 0.341i)T \)
7 \( 1 + (2.06 + 1.64i)T \)
good11 \( 1 - 1.06iT - 11T^{2} \)
13 \( 1 - 4.82T + 13T^{2} \)
17 \( 1 + 7.89iT - 17T^{2} \)
19 \( 1 + 4.02iT - 19T^{2} \)
23 \( 1 + 5.69T + 23T^{2} \)
29 \( 1 + 2.00iT - 29T^{2} \)
31 \( 1 + 4.89iT - 31T^{2} \)
37 \( 1 + 2.56iT - 37T^{2} \)
41 \( 1 + 5.08T + 41T^{2} \)
43 \( 1 - 6.15iT - 43T^{2} \)
47 \( 1 + 2.27iT - 47T^{2} \)
53 \( 1 + 9.84T + 53T^{2} \)
59 \( 1 - 5.87T + 59T^{2} \)
61 \( 1 + 7.02iT - 61T^{2} \)
67 \( 1 + 10.8iT - 67T^{2} \)
71 \( 1 - 0.0512iT - 71T^{2} \)
73 \( 1 - 2.86T + 73T^{2} \)
79 \( 1 - 7.00T + 79T^{2} \)
83 \( 1 - 7.59iT - 83T^{2} \)
89 \( 1 + 9.72T + 89T^{2} \)
97 \( 1 + 1.77T + 97T^{2} \)
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

−9.842688090694737865311522644445, −9.292682903990387071172990924955, −8.100611410587731971799439323233, −7.10888390151934142939818925470, −6.50411277018066058634238342216, −5.80885335069923110539001049525, −4.72651635306737003974927634450, −3.19814966971782166190161172057, −2.09792066551448091972897163846, −0.67914696638968824012497733976, 1.65626557488323893893597210074, 3.25547123066348159162119503177, 4.01924448428453751400661956590, 5.45521341278857099196058654851, 6.02472925785031378128680318513, 6.47059470762349153164536873687, 8.487601962564042413241626388509, 8.690674091496223795626057608602, 9.872098464309186315969287678881, 10.27846631779570670466420470132

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