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Properties

Label 2-840-21.5-c1-0-27
Degree $2$
Conductor $840$
Sign $-0.613 + 0.789i$
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  + (−0.645 − 1.60i)3-s + (0.5 − 0.866i)5-s + (2.09 − 1.61i)7-s + (−2.16 + 2.07i)9-s + (0.439 − 0.253i)11-s − 3.23i·13-s + (−1.71 − 0.244i)15-s + (−3.31 − 5.73i)17-s + (6.66 + 3.84i)19-s + (−3.94 − 2.33i)21-s + (−0.899 − 0.519i)23-s + (−0.499 − 0.866i)25-s + (4.73 + 2.14i)27-s − 1.87i·29-s + (−3.00 + 1.73i)31-s + ⋯
L(s)  = 1  + (−0.372 − 0.927i)3-s + (0.223 − 0.387i)5-s + (0.793 − 0.608i)7-s + (−0.722 + 0.691i)9-s + (0.132 − 0.0765i)11-s − 0.897i·13-s + (−0.442 − 0.0631i)15-s + (−0.803 − 1.39i)17-s + (1.52 + 0.882i)19-s + (−0.860 − 0.509i)21-s + (−0.187 − 0.108i)23-s + (−0.0999 − 0.173i)25-s + (0.910 + 0.412i)27-s − 0.348i·29-s + (−0.539 + 0.311i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.613 + 0.789i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (761, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.613 + 0.789i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.591877 - 1.21024i\)
\(L(\frac12)\) \(\approx\) \(0.591877 - 1.21024i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.645 + 1.60i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.09 + 1.61i)T \)
good11 \( 1 + (-0.439 + 0.253i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.23iT - 13T^{2} \)
17 \( 1 + (3.31 + 5.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.66 - 3.84i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.899 + 0.519i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 1.87iT - 29T^{2} \)
31 \( 1 + (3.00 - 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.777 + 1.34i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.119T + 41T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + (-2.36 + 4.09i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.82 - 5.09i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.706 + 1.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (8.40 + 4.85i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.88 - 4.99i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.05iT - 71T^{2} \)
73 \( 1 + (-8.86 + 5.11i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.36 + 4.09i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.07T + 83T^{2} \)
89 \( 1 + (4.41 - 7.63i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.08iT - 97T^{2} \)
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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

−9.968504608544967863055530357453, −8.951436050193784705588259535463, −7.908749050716771718360424437193, −7.50681943194219959803616050657, −6.48404136855828571763547811119, −5.40014036248110292395455381908, −4.83251358296724461238343865503, −3.26637015559634750091923118112, −1.86213267715008817861107973396, −0.71047426290069917178498678753, 1.80825597272515342252545174986, 3.18439935737929716845377714032, 4.32295204496624982505935404406, 5.12574492037336050561961964476, 6.02919151998023486954318757832, 6.89868747277368052359774902946, 8.145215474297775152510232143152, 9.025065083083383030075691736775, 9.590992781544921082273817990436, 10.58291930035595011884085202028

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