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Properties

Label 2-840-840.293-c0-0-5
Degree $2$
Conductor $840$
Sign $0.229 + 0.973i$
Analytic cond. $0.419214$
Root an. cond. $0.647467$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)2-s + (0.382 − 0.923i)3-s + 1.00i·4-s + (−0.382 + 0.923i)5-s + (−0.923 + 0.382i)6-s + (0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.923 − 0.382i)10-s + (0.923 + 0.382i)12-s + (1.30 + 1.30i)13-s − 1.00·14-s + (0.707 + 0.707i)15-s − 1.00·16-s + i·18-s − 0.765i·19-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (0.382 − 0.923i)3-s + 1.00i·4-s + (−0.382 + 0.923i)5-s + (−0.923 + 0.382i)6-s + (0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (0.923 − 0.382i)10-s + (0.923 + 0.382i)12-s + (1.30 + 1.30i)13-s − 1.00·14-s + (0.707 + 0.707i)15-s − 1.00·16-s + i·18-s − 0.765i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.229 + 0.973i$
Analytic conductor: \(0.419214\)
Root analytic conductor: \(0.647467\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :0),\ 0.229 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8127140362\)
\(L(\frac12)\) \(\approx\) \(0.8127140362\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
good11 \( 1 + T^{2} \)
13 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + 0.765iT - T^{2} \)
23 \( 1 + (-1 + i)T - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + 0.765T + T^{2} \)
61 \( 1 + 1.84T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−10.47273894365546179990825243489, −9.191722695227310098641564305698, −8.563429567315471317763893097629, −7.74239952910382920872685612701, −6.97289958916690344808545960582, −6.45289456093010333038197851981, −4.42476091110298767738284224309, −3.50255551713880824964302553645, −2.43594433320214708286483971695, −1.26079521219839814510303856886, 1.50985470321534075997042451947, 3.29815835664180795271384971436, 4.57950626846285614279986335162, 5.39869069388467044589086831587, 5.94383173349934888519950519421, 7.75373326226670262663843495521, 8.099807878916217670766129894382, 8.910482831749082906613586521224, 9.350918827467259465867299141608, 10.51259310371125956500564851379

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