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Properties

Label 2-840-105.104-c1-0-35
Degree $2$
Conductor $840$
Sign $0.893 + 0.448i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.70 − 0.301i)3-s + (2.04 − 0.902i)5-s + (−1.30 − 2.30i)7-s + (2.81 − 1.02i)9-s + 5.71i·11-s + 3.76·13-s + (3.21 − 2.15i)15-s + 3.22i·17-s − 0.786i·19-s + (−2.91 − 3.53i)21-s − 1.52·23-s + (3.37 − 3.69i)25-s + (4.49 − 2.60i)27-s − 6.77i·29-s + 1.56i·31-s + ⋯
L(s)  = 1  + (0.984 − 0.174i)3-s + (0.914 − 0.403i)5-s + (−0.492 − 0.870i)7-s + (0.939 − 0.343i)9-s + 1.72i·11-s + 1.04·13-s + (0.830 − 0.556i)15-s + 0.783i·17-s − 0.180i·19-s + (−0.636 − 0.771i)21-s − 0.318·23-s + (0.674 − 0.738i)25-s + (0.865 − 0.501i)27-s − 1.25i·29-s + 0.281i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.893 + 0.448i$
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: \((2,\ 840,\ (\ :1/2),\ 0.893 + 0.448i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.47346 - 0.586344i\)
\(L(\frac12)\) \(\approx\) \(2.47346 - 0.586344i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 0.301i)T \)
5 \( 1 + (-2.04 + 0.902i)T \)
7 \( 1 + (1.30 + 2.30i)T \)
good11 \( 1 - 5.71iT - 11T^{2} \)
13 \( 1 - 3.76T + 13T^{2} \)
17 \( 1 - 3.22iT - 17T^{2} \)
19 \( 1 + 0.786iT - 19T^{2} \)
23 \( 1 + 1.52T + 23T^{2} \)
29 \( 1 + 6.77iT - 29T^{2} \)
31 \( 1 - 1.56iT - 31T^{2} \)
37 \( 1 + 8.83iT - 37T^{2} \)
41 \( 1 + 6.65T + 41T^{2} \)
43 \( 1 + 8.85iT - 43T^{2} \)
47 \( 1 - 1.75iT - 47T^{2} \)
53 \( 1 + 0.616T + 53T^{2} \)
59 \( 1 + 6.38T + 59T^{2} \)
61 \( 1 - 14.4iT - 61T^{2} \)
67 \( 1 - 6.26iT - 67T^{2} \)
71 \( 1 - 5.09iT - 71T^{2} \)
73 \( 1 + 12.5T + 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 - 10.5iT - 83T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 - 7.69T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.12898656170417346571588232994, −9.299444355047694451625052517174, −8.586894737173491020373928016338, −7.53732214830584209205793527687, −6.83607566073383995849941564303, −5.90254538891095556837283805107, −4.47226158987608575496617905493, −3.79965413885540508032653698607, −2.34911474052140072272163226365, −1.40431540948461569631166864916, 1.60061199285724992866263870387, 3.03913081729102990736137645141, 3.28983081379479919657200966281, 5.01567183242677669635750955668, 6.04139496913786541067973330697, 6.61287856189239542243626618304, 7.987662933716208115300287727662, 8.763999073119274599640582967604, 9.237388689908146351962955455068, 10.12881441706821441469259275461

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