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Properties

Label 2-840-35.13-c1-0-14
Degree $2$
Conductor $840$
Sign $0.995 + 0.0901i$
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 + (1.35 + 1.77i)5-s + (2.59 + 0.498i)7-s − 1.00i·9-s − 1.97·11-s + (4.59 − 4.59i)13-s + (2.21 + 0.295i)15-s + (−4.07 − 4.07i)17-s + 0.683·19-s + (2.18 − 1.48i)21-s + (5.64 + 5.64i)23-s + (−1.31 + 4.82i)25-s + (−0.707 − 0.707i)27-s + 5.55i·29-s + 6.08i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.607 + 0.794i)5-s + (0.982 + 0.188i)7-s − 0.333i·9-s − 0.594·11-s + (1.27 − 1.27i)13-s + (0.572 + 0.0763i)15-s + (−0.988 − 0.988i)17-s + 0.156·19-s + (0.477 − 0.324i)21-s + (1.17 + 1.17i)23-s + (−0.262 + 0.964i)25-s + (−0.136 − 0.136i)27-s + 1.03i·29-s + 1.09i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.995 + 0.0901i$
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: \((2,\ 840,\ (\ :1/2),\ 0.995 + 0.0901i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21543 - 0.100031i\)
\(L(\frac12)\) \(\approx\) \(2.21543 - 0.100031i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.35 - 1.77i)T \)
7 \( 1 + (-2.59 - 0.498i)T \)
good11 \( 1 + 1.97T + 11T^{2} \)
13 \( 1 + (-4.59 + 4.59i)T - 13iT^{2} \)
17 \( 1 + (4.07 + 4.07i)T + 17iT^{2} \)
19 \( 1 - 0.683T + 19T^{2} \)
23 \( 1 + (-5.64 - 5.64i)T + 23iT^{2} \)
29 \( 1 - 5.55iT - 29T^{2} \)
31 \( 1 - 6.08iT - 31T^{2} \)
37 \( 1 + (-2.16 + 2.16i)T - 37iT^{2} \)
41 \( 1 + 9.16iT - 41T^{2} \)
43 \( 1 + (-0.140 - 0.140i)T + 43iT^{2} \)
47 \( 1 + (1.75 + 1.75i)T + 47iT^{2} \)
53 \( 1 + (0.681 + 0.681i)T + 53iT^{2} \)
59 \( 1 - 9.12T + 59T^{2} \)
61 \( 1 + 2.91iT - 61T^{2} \)
67 \( 1 + (7.72 - 7.72i)T - 67iT^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 + (-7.15 + 7.15i)T - 73iT^{2} \)
79 \( 1 + 12.3iT - 79T^{2} \)
83 \( 1 + (7.79 - 7.79i)T - 83iT^{2} \)
89 \( 1 + 9.74T + 89T^{2} \)
97 \( 1 + (0.937 + 0.937i)T + 97iT^{2} \)
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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

−10.40558251191308082154432121896, −9.138562529940985672756721498696, −8.566613631650980915802743106349, −7.52346505259569115019532851648, −6.95095833231535083389830037838, −5.71854333441138169055879296496, −5.09511476662282204019046460396, −3.44492846083330308343744087736, −2.61869453228522833448432594010, −1.37313776903999668560137751474, 1.39940145703834690666881895139, 2.45089540033076319425602453436, 4.21937621026917917252756468043, 4.55700107526890098494616079526, 5.77282971208282461866649086990, 6.65250431778228288298663820973, 8.055604594628012554979943976205, 8.550112676982733258388780890239, 9.211724938992926241517116771088, 10.17437777881337723035489740260

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