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Properties

Label 2-840-280.139-c1-0-82
Degree $2$
Conductor $840$
Sign $0.995 - 0.0997i$
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  + (1.38 + 0.273i)2-s + 3-s + (1.85 + 0.758i)4-s + (0.840 − 2.07i)5-s + (1.38 + 0.273i)6-s + (2.31 + 1.27i)7-s + (2.36 + 1.55i)8-s + 9-s + (1.73 − 2.64i)10-s − 1.15·11-s + (1.85 + 0.758i)12-s + 0.698i·13-s + (2.87 + 2.39i)14-s + (0.840 − 2.07i)15-s + (2.84 + 2.80i)16-s − 6.87·17-s + ⋯
L(s)  = 1  + (0.981 + 0.193i)2-s + 0.577·3-s + (0.925 + 0.379i)4-s + (0.376 − 0.926i)5-s + (0.566 + 0.111i)6-s + (0.876 + 0.480i)7-s + (0.834 + 0.551i)8-s + 0.333·9-s + (0.548 − 0.836i)10-s − 0.349·11-s + (0.534 + 0.219i)12-s + 0.193i·13-s + (0.767 + 0.641i)14-s + (0.217 − 0.534i)15-s + (0.712 + 0.702i)16-s − 1.66·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.995 - 0.0997i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.995 - 0.0997i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.83094 + 0.191525i\)
\(L(\frac12)\) \(\approx\) \(3.83094 + 0.191525i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 - 0.273i)T \)
3 \( 1 - T \)
5 \( 1 + (-0.840 + 2.07i)T \)
7 \( 1 + (-2.31 - 1.27i)T \)
good11 \( 1 + 1.15T + 11T^{2} \)
13 \( 1 - 0.698iT - 13T^{2} \)
17 \( 1 + 6.87T + 17T^{2} \)
19 \( 1 + 4.94iT - 19T^{2} \)
23 \( 1 + 5.30T + 23T^{2} \)
29 \( 1 - 10.0iT - 29T^{2} \)
31 \( 1 + 0.109T + 31T^{2} \)
37 \( 1 - 4.30T + 37T^{2} \)
41 \( 1 + 2.58iT - 41T^{2} \)
43 \( 1 + 7.55iT - 43T^{2} \)
47 \( 1 + 3.53iT - 47T^{2} \)
53 \( 1 + 1.70T + 53T^{2} \)
59 \( 1 - 12.3iT - 59T^{2} \)
61 \( 1 - 4.88T + 61T^{2} \)
67 \( 1 + 8.12iT - 67T^{2} \)
71 \( 1 - 5.80iT - 71T^{2} \)
73 \( 1 + 3.86T + 73T^{2} \)
79 \( 1 + 6.30iT - 79T^{2} \)
83 \( 1 + 12.7T + 83T^{2} \)
89 \( 1 + 0.841iT - 89T^{2} \)
97 \( 1 + 0.0488T + 97T^{2} \)
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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

−10.36490131644828480162726111679, −8.944502271905850676816528163627, −8.665662036550675569488047650715, −7.61694266694347634813117432827, −6.68668467809253319603139561568, −5.55892653059889190034114295742, −4.81244221124210708902796156168, −4.12091108110476890244935416615, −2.57466053411741518419984293654, −1.78717203892831401564312569180, 1.84691977229868542632564236925, 2.60146873091824569086866462797, 3.86258406022660435192636942132, 4.55086596529255090016660040406, 5.85935768300977345045771810188, 6.55923438807261302343873151901, 7.62735159354098999294475391330, 8.147229505805804238555791440042, 9.721387884593579409685344119183, 10.27347053153891935953884412088

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