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Properties

Label 2-840-105.59-c1-0-25
Degree $2$
Conductor $840$
Sign $0.919 + 0.394i$
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  + (−0.381 − 1.68i)3-s + (1.27 + 1.83i)5-s + (1.96 − 1.77i)7-s + (−2.70 + 1.29i)9-s + (3.95 + 2.28i)11-s − 1.57·13-s + (2.62 − 2.85i)15-s + (4.06 + 2.34i)17-s + (0.0201 − 0.0116i)19-s + (−3.74 − 2.63i)21-s + (−0.0964 − 0.166i)23-s + (−1.76 + 4.67i)25-s + (3.21 + 4.08i)27-s − 4.98i·29-s + (5.92 + 3.42i)31-s + ⋯
L(s)  = 1  + (−0.220 − 0.975i)3-s + (0.569 + 0.822i)5-s + (0.741 − 0.670i)7-s + (−0.902 + 0.430i)9-s + (1.19 + 0.689i)11-s − 0.436·13-s + (0.676 − 0.736i)15-s + (0.986 + 0.569i)17-s + (0.00461 − 0.00266i)19-s + (−0.817 − 0.575i)21-s + (−0.0201 − 0.0348i)23-s + (−0.352 + 0.935i)25-s + (0.618 + 0.785i)27-s − 0.925i·29-s + (1.06 + 0.614i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.919 + 0.394i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.919 + 0.394i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76494 - 0.362457i\)
\(L(\frac12)\) \(\approx\) \(1.76494 - 0.362457i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.381 + 1.68i)T \)
5 \( 1 + (-1.27 - 1.83i)T \)
7 \( 1 + (-1.96 + 1.77i)T \)
good11 \( 1 + (-3.95 - 2.28i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.57T + 13T^{2} \)
17 \( 1 + (-4.06 - 2.34i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.0201 + 0.0116i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0964 + 0.166i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.98iT - 29T^{2} \)
31 \( 1 + (-5.92 - 3.42i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.79 + 1.03i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.85T + 41T^{2} \)
43 \( 1 + 6.60iT - 43T^{2} \)
47 \( 1 + (-4.93 + 2.84i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.13 - 8.90i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.24 - 2.16i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.02 + 4.63i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.87 + 3.39i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.0iT - 71T^{2} \)
73 \( 1 + (-7.23 + 12.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.60 - 11.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.02iT - 83T^{2} \)
89 \( 1 + (0.594 + 1.03i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.721T + 97T^{2} \)
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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

−10.28077050729277159914764142606, −9.392014725666088672317252207777, −8.190741765946345610379728579652, −7.44263174401612887070005299121, −6.75672036475076277296873024978, −6.02568981271095203897603009573, −4.90712106379291985921406974196, −3.62892485528357837445313200955, −2.23657177677027978176048803814, −1.28099934008640552093495235683, 1.18855902427885091173688147521, 2.79461375033714514130810815965, 4.07766537334646237850464765693, 5.02164524824098919969758986995, 5.59672496260012552515608336910, 6.51157286564938348751885645802, 8.079101267866127909320992731353, 8.725692627947698991342529692619, 9.463541575298834271734963177012, 10.00516915660851195379016575142

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