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Properties

Label 2-840-21.17-c1-0-22
Degree $2$
Conductor $840$
Sign $0.993 + 0.115i$
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 + 0.324i)3-s + (0.5 + 0.866i)5-s + (0.143 − 2.64i)7-s + (2.78 + 1.10i)9-s + (2.62 + 1.51i)11-s − 6.46i·13-s + (0.569 + 1.63i)15-s + (−0.925 + 1.60i)17-s + (3.98 − 2.30i)19-s + (1.10 − 4.44i)21-s + (−6.74 + 3.89i)23-s + (−0.499 + 0.866i)25-s + (4.38 + 2.78i)27-s − 3.03i·29-s + (4.02 + 2.32i)31-s + ⋯
L(s)  = 1  + (0.982 + 0.187i)3-s + (0.223 + 0.387i)5-s + (0.0543 − 0.998i)7-s + (0.929 + 0.368i)9-s + (0.790 + 0.456i)11-s − 1.79i·13-s + (0.147 + 0.422i)15-s + (−0.224 + 0.388i)17-s + (0.914 − 0.527i)19-s + (0.240 − 0.970i)21-s + (−1.40 + 0.812i)23-s + (−0.0999 + 0.173i)25-s + (0.844 + 0.535i)27-s − 0.563i·29-s + (0.723 + 0.417i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.993 + 0.115i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.993 + 0.115i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.40392 - 0.139765i\)
\(L(\frac12)\) \(\approx\) \(2.40392 - 0.139765i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 - 0.324i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-0.143 + 2.64i)T \)
good11 \( 1 + (-2.62 - 1.51i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 6.46iT - 13T^{2} \)
17 \( 1 + (0.925 - 1.60i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.98 + 2.30i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.74 - 3.89i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.03iT - 29T^{2} \)
31 \( 1 + (-4.02 - 2.32i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.56 - 4.43i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 9.52T + 41T^{2} \)
43 \( 1 + 6.53T + 43T^{2} \)
47 \( 1 + (1.70 + 2.95i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.06 - 0.612i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.40 - 11.0i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.46 - 1.42i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.02 + 1.78i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.91iT - 71T^{2} \)
73 \( 1 + (-7.84 - 4.52i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.79 + 3.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + (8.32 + 14.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.92iT - 97T^{2} \)
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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

−9.999266473361904395269524843349, −9.611622151837146078396829482657, −8.338078405454255438605771068954, −7.70589804950270872784867750084, −6.99379282964190604739659162825, −5.85373650745892326817100895365, −4.55450602536834480768859162179, −3.66788249071617645124121507578, −2.77808470881449250175954799844, −1.30847196783126972023407987085, 1.56708326224526795219977184150, 2.48863487558468647429114548302, 3.79287745764824790762275646071, 4.66856415025779272855172679626, 6.03372975519594870432430672040, 6.70876499861890948128280046072, 7.898773920077694223216440185233, 8.631780511678739436851928538983, 9.385153971227461596986766888726, 9.673471025488888181269254945314

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