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Properties

Label 2-840-56.27-c1-0-46
Degree $2$
Conductor $840$
Sign $0.887 - 0.461i$
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  + (1.41 + 0.0894i)2-s + i·3-s + (1.98 + 0.252i)4-s + 5-s + (−0.0894 + 1.41i)6-s + (−1.64 − 2.07i)7-s + (2.77 + 0.533i)8-s − 9-s + (1.41 + 0.0894i)10-s + 3.15·11-s + (−0.252 + 1.98i)12-s + 2.45·13-s + (−2.13 − 3.07i)14-s + i·15-s + (3.87 + 1.00i)16-s + 2.54i·17-s + ⋯
L(s)  = 1  + (0.997 + 0.0632i)2-s + 0.577i·3-s + (0.991 + 0.126i)4-s + 0.447·5-s + (−0.0365 + 0.576i)6-s + (−0.620 − 0.784i)7-s + (0.982 + 0.188i)8-s − 0.333·9-s + (0.446 + 0.0282i)10-s + 0.952·11-s + (−0.0728 + 0.572i)12-s + 0.680·13-s + (−0.569 − 0.821i)14-s + 0.258i·15-s + (0.968 + 0.250i)16-s + 0.616i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 - 0.461i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.887 - 0.461i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.887 - 0.461i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.10548 + 0.758880i\)
\(L(\frac12)\) \(\approx\) \(3.10548 + 0.758880i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0894i)T \)
3 \( 1 - iT \)
5 \( 1 - T \)
7 \( 1 + (1.64 + 2.07i)T \)
good11 \( 1 - 3.15T + 11T^{2} \)
13 \( 1 - 2.45T + 13T^{2} \)
17 \( 1 - 2.54iT - 17T^{2} \)
19 \( 1 - 0.844iT - 19T^{2} \)
23 \( 1 + 1.41iT - 23T^{2} \)
29 \( 1 - 4.32iT - 29T^{2} \)
31 \( 1 + 0.885T + 31T^{2} \)
37 \( 1 + 6.27iT - 37T^{2} \)
41 \( 1 - 0.616iT - 41T^{2} \)
43 \( 1 + 4.29T + 43T^{2} \)
47 \( 1 + 6.73T + 47T^{2} \)
53 \( 1 - 6.93iT - 53T^{2} \)
59 \( 1 + 11.9iT - 59T^{2} \)
61 \( 1 + 5.56T + 61T^{2} \)
67 \( 1 + 14.8T + 67T^{2} \)
71 \( 1 + 3.33iT - 71T^{2} \)
73 \( 1 - 0.330iT - 73T^{2} \)
79 \( 1 - 8.66iT - 79T^{2} \)
83 \( 1 + 17.0iT - 83T^{2} \)
89 \( 1 + 2.25iT - 89T^{2} \)
97 \( 1 + 2.37iT - 97T^{2} \)
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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

−10.49226207082987842088447114535, −9.587276435468338893415354605110, −8.643565134504487734832325611954, −7.44337913797163498788532200041, −6.47091091990447650624012137088, −5.97178836656902785703853229618, −4.76188847751186264304352289422, −3.87217622405506945424544505568, −3.20367159938163688916247612601, −1.57792331053764263845683301432, 1.48031939877408587409655123579, 2.65365807424470644593815046703, 3.58783022548098686981394507103, 4.87813414251406128079550455950, 5.93011550461683221836873598067, 6.39332916052801530172223526455, 7.21652823975941398803451361462, 8.415969673741896766184241558146, 9.335681687861853259839023565542, 10.18794833846853332524742843613

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