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Properties

Label 2-840-280.237-c1-0-38
Degree $2$
Conductor $840$
Sign $0.955 + 0.294i$
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  + (−1.08 + 0.904i)2-s + (−0.707 + 0.707i)3-s + (0.364 − 1.96i)4-s + (−0.998 + 2.00i)5-s + (0.129 − 1.40i)6-s + (−2.62 − 0.301i)7-s + (1.38 + 2.46i)8-s − 1.00i·9-s + (−0.723 − 3.07i)10-s − 2.31i·11-s + (1.13 + 1.64i)12-s + (−3.26 + 3.26i)13-s + (3.13 − 2.04i)14-s + (−0.708 − 2.12i)15-s + (−3.73 − 1.43i)16-s + (2.54 − 2.54i)17-s + ⋯
L(s)  = 1  + (−0.768 + 0.639i)2-s + (−0.408 + 0.408i)3-s + (0.182 − 0.983i)4-s + (−0.446 + 0.894i)5-s + (0.0527 − 0.574i)6-s + (−0.993 − 0.113i)7-s + (0.488 + 0.872i)8-s − 0.333i·9-s + (−0.228 − 0.973i)10-s − 0.697i·11-s + (0.327 + 0.475i)12-s + (−0.905 + 0.905i)13-s + (0.836 − 0.547i)14-s + (−0.182 − 0.547i)15-s + (−0.933 − 0.358i)16-s + (0.616 − 0.616i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 + 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.955 + 0.294i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.955 + 0.294i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.407294 - 0.0612387i\)
\(L(\frac12)\) \(\approx\) \(0.407294 - 0.0612387i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 - 0.904i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.998 - 2.00i)T \)
7 \( 1 + (2.62 + 0.301i)T \)
good11 \( 1 + 2.31iT - 11T^{2} \)
13 \( 1 + (3.26 - 3.26i)T - 13iT^{2} \)
17 \( 1 + (-2.54 + 2.54i)T - 17iT^{2} \)
19 \( 1 + 2.46iT - 19T^{2} \)
23 \( 1 + (2.19 - 2.19i)T - 23iT^{2} \)
29 \( 1 - 5.31T + 29T^{2} \)
31 \( 1 + 0.162iT - 31T^{2} \)
37 \( 1 + (0.503 - 0.503i)T - 37iT^{2} \)
41 \( 1 + 0.552iT - 41T^{2} \)
43 \( 1 + (5.38 + 5.38i)T + 43iT^{2} \)
47 \( 1 + (-9.44 + 9.44i)T - 47iT^{2} \)
53 \( 1 + (-2.25 - 2.25i)T + 53iT^{2} \)
59 \( 1 + 0.0510iT - 59T^{2} \)
61 \( 1 + 4.19T + 61T^{2} \)
67 \( 1 + (-5.01 + 5.01i)T - 67iT^{2} \)
71 \( 1 - 14.1T + 71T^{2} \)
73 \( 1 + (-7.08 - 7.08i)T + 73iT^{2} \)
79 \( 1 + 16.6iT - 79T^{2} \)
83 \( 1 + (-0.691 + 0.691i)T - 83iT^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 + (1.80 - 1.80i)T - 97iT^{2} \)
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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

−10.03488152687153880897288268923, −9.471818179888510851693109643028, −8.523381051585743138863752362145, −7.34947079503130132306942878467, −6.84812607692469450249616796955, −6.06657787345295975477567072860, −5.03838377437609504201855286072, −3.75247959137520703159312487330, −2.55346717355752030214276263609, −0.34235357194183162307081458252, 0.994950300498789841991209068042, 2.45726890358848926591198555318, 3.67086663103021275378089632659, 4.77493030294767836222412989580, 5.96857871627968027446020652470, 7.06038202608279338491660467083, 7.87637687160076862515029502675, 8.489024439226068946759815149026, 9.702754773886603777456343424358, 9.998659762904731226910424819021

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