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Properties

Label 2-840-24.11-c1-0-74
Degree $2$
Conductor $840$
Sign $-0.873 + 0.486i$
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  + (−1.15 − 0.822i)2-s + (1.20 − 1.24i)3-s + (0.648 + 1.89i)4-s − 5-s + (−2.41 + 0.434i)6-s + i·7-s + (0.809 − 2.71i)8-s + (−0.0783 − 2.99i)9-s + (1.15 + 0.822i)10-s − 3.79i·11-s + (3.13 + 1.48i)12-s − 5.34i·13-s + (0.822 − 1.15i)14-s + (−1.20 + 1.24i)15-s + (−3.15 + 2.45i)16-s + 7.31i·17-s + ⋯
L(s)  = 1  + (−0.813 − 0.581i)2-s + (0.697 − 0.716i)3-s + (0.324 + 0.945i)4-s − 0.447·5-s + (−0.984 + 0.177i)6-s + 0.377i·7-s + (0.286 − 0.958i)8-s + (−0.0261 − 0.999i)9-s + (0.363 + 0.259i)10-s − 1.14i·11-s + (0.903 + 0.427i)12-s − 1.48i·13-s + (0.219 − 0.307i)14-s + (−0.312 + 0.320i)15-s + (−0.789 + 0.613i)16-s + 1.77i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.873 + 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.873 + 0.486i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.873 + 0.486i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.247020 - 0.950796i\)
\(L(\frac12)\) \(\approx\) \(0.247020 - 0.950796i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 + 0.822i)T \)
3 \( 1 + (-1.20 + 1.24i)T \)
5 \( 1 + T \)
7 \( 1 - iT \)
good11 \( 1 + 3.79iT - 11T^{2} \)
13 \( 1 + 5.34iT - 13T^{2} \)
17 \( 1 - 7.31iT - 17T^{2} \)
19 \( 1 - 1.37T + 19T^{2} \)
23 \( 1 - 1.57T + 23T^{2} \)
29 \( 1 + 2.35T + 29T^{2} \)
31 \( 1 + 5.51iT - 31T^{2} \)
37 \( 1 + 4.62iT - 37T^{2} \)
41 \( 1 + 11.5iT - 41T^{2} \)
43 \( 1 + 4.45T + 43T^{2} \)
47 \( 1 + 9.40T + 47T^{2} \)
53 \( 1 + 9.32T + 53T^{2} \)
59 \( 1 + 6.51iT - 59T^{2} \)
61 \( 1 + 1.39iT - 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 8.52T + 73T^{2} \)
79 \( 1 + 1.16iT - 79T^{2} \)
83 \( 1 - 4.24iT - 83T^{2} \)
89 \( 1 - 11.7iT - 89T^{2} \)
97 \( 1 - 7.43T + 97T^{2} \)
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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

−9.747332473335483644239427043551, −8.806245585264581224108142200764, −8.145394683123557454129688972019, −7.81678850765989673068061199173, −6.60793866982497061298623599341, −5.66640694709983805925222317732, −3.72294554044341855839326985999, −3.21190901952904051832216113971, −1.97514694086112762150791572528, −0.57685124587121063208423059259, 1.71116948938464412414557877087, 3.05623743525490554703281453550, 4.59995342693865401821691505197, 4.91826728160574223344600292584, 6.68361670179663529136781841563, 7.24385845936256594732411245084, 8.030395572114979211255113258627, 9.016660438289947393051102873587, 9.596393807912756742319810601003, 10.10474987190862534856762438599

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