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Properties

Label 2-840-35.13-c1-0-19
Degree $2$
Conductor $840$
Sign $-0.242 + 0.970i$
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  + (0.707 − 0.707i)3-s + (−1.43 + 1.71i)5-s + (−2.23 − 1.41i)7-s − 1.00i·9-s + 0.566·11-s + (5.03 − 5.03i)13-s + (0.194 + 2.22i)15-s + (−0.984 − 0.984i)17-s − 7.61·19-s + (−2.58 + 0.581i)21-s + (2.55 + 2.55i)23-s + (−0.865 − 4.92i)25-s + (−0.707 − 0.707i)27-s − 7.85i·29-s − 7.09i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.642 + 0.765i)5-s + (−0.845 − 0.534i)7-s − 0.333i·9-s + 0.170·11-s + (1.39 − 1.39i)13-s + (0.0501 + 0.575i)15-s + (−0.238 − 0.238i)17-s − 1.74·19-s + (−0.563 + 0.126i)21-s + (0.531 + 0.531i)23-s + (−0.173 − 0.984i)25-s + (−0.136 − 0.136i)27-s − 1.45i·29-s − 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.242 + 0.970i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.675333 - 0.865098i\)
\(L(\frac12)\) \(\approx\) \(0.675333 - 0.865098i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (1.43 - 1.71i)T \)
7 \( 1 + (2.23 + 1.41i)T \)
good11 \( 1 - 0.566T + 11T^{2} \)
13 \( 1 + (-5.03 + 5.03i)T - 13iT^{2} \)
17 \( 1 + (0.984 + 0.984i)T + 17iT^{2} \)
19 \( 1 + 7.61T + 19T^{2} \)
23 \( 1 + (-2.55 - 2.55i)T + 23iT^{2} \)
29 \( 1 + 7.85iT - 29T^{2} \)
31 \( 1 + 7.09iT - 31T^{2} \)
37 \( 1 + (-0.887 + 0.887i)T - 37iT^{2} \)
41 \( 1 + 6.29iT - 41T^{2} \)
43 \( 1 + (2.74 + 2.74i)T + 43iT^{2} \)
47 \( 1 + (3.25 + 3.25i)T + 47iT^{2} \)
53 \( 1 + (-7.27 - 7.27i)T + 53iT^{2} \)
59 \( 1 + 6.62T + 59T^{2} \)
61 \( 1 + 5.46iT - 61T^{2} \)
67 \( 1 + (2.53 - 2.53i)T - 67iT^{2} \)
71 \( 1 - 8.01T + 71T^{2} \)
73 \( 1 + (8.97 - 8.97i)T - 73iT^{2} \)
79 \( 1 + 1.85iT - 79T^{2} \)
83 \( 1 + (-2.61 + 2.61i)T - 83iT^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 + (-4.32 - 4.32i)T + 97iT^{2} \)
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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

−10.13465235863673446542568369041, −8.983193662407812200966122043007, −8.141641071269794780835107677741, −7.45136669811487981789554383910, −6.50292487150766693118016928242, −5.93597310899152871379391474561, −4.10819579060974605066453361679, −3.50816414617626958014314105307, −2.45148370786373822793847807644, −0.51389453107769032816713804243, 1.65287513827098090009000566397, 3.20952347962959226949314268150, 4.08258096188131939148406173414, 4.87615689803654401554947486743, 6.25275614978906536832984260905, 6.84211341544115944947116329185, 8.347361867993989534873272132224, 8.795531169727764635301472785043, 9.243678106717291783606772104607, 10.48650324070216049939624592379

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