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Properties

Label 2-840-280.139-c1-0-5
Degree $2$
Conductor $840$
Sign $-0.726 + 0.686i$
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  + (0.752 + 1.19i)2-s + 3-s + (−0.867 + 1.80i)4-s + (−1.64 + 1.51i)5-s + (0.752 + 1.19i)6-s + (−2.63 − 0.259i)7-s + (−2.81 + 0.318i)8-s + 9-s + (−3.05 − 0.833i)10-s − 3.15·11-s + (−0.867 + 1.80i)12-s − 2.42i·13-s + (−1.67 − 3.34i)14-s + (−1.64 + 1.51i)15-s + (−2.49 − 3.12i)16-s + 2.13·17-s + ⋯
L(s)  = 1  + (0.532 + 0.846i)2-s + 0.577·3-s + (−0.433 + 0.901i)4-s + (−0.736 + 0.676i)5-s + (0.307 + 0.488i)6-s + (−0.995 − 0.0980i)7-s + (−0.993 + 0.112i)8-s + 0.333·9-s + (−0.964 − 0.263i)10-s − 0.952·11-s + (−0.250 + 0.520i)12-s − 0.673i·13-s + (−0.446 − 0.894i)14-s + (−0.425 + 0.390i)15-s + (−0.623 − 0.781i)16-s + 0.517·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 + 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.726 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.726 + 0.686i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ -0.726 + 0.686i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.255678 - 0.642637i\)
\(L(\frac12)\) \(\approx\) \(0.255678 - 0.642637i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.752 - 1.19i)T \)
3 \( 1 - T \)
5 \( 1 + (1.64 - 1.51i)T \)
7 \( 1 + (2.63 + 0.259i)T \)
good11 \( 1 + 3.15T + 11T^{2} \)
13 \( 1 + 2.42iT - 13T^{2} \)
17 \( 1 - 2.13T + 17T^{2} \)
19 \( 1 - 2.84iT - 19T^{2} \)
23 \( 1 + 7.13T + 23T^{2} \)
29 \( 1 - 4.57iT - 29T^{2} \)
31 \( 1 - 3.50T + 31T^{2} \)
37 \( 1 + 7.05T + 37T^{2} \)
41 \( 1 + 2.43iT - 41T^{2} \)
43 \( 1 - 5.07iT - 43T^{2} \)
47 \( 1 - 12.8iT - 47T^{2} \)
53 \( 1 + 4.91T + 53T^{2} \)
59 \( 1 - 2.93iT - 59T^{2} \)
61 \( 1 - 1.17T + 61T^{2} \)
67 \( 1 - 5.88iT - 67T^{2} \)
71 \( 1 - 5.30iT - 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 + 4.53iT - 79T^{2} \)
83 \( 1 + 14.0T + 83T^{2} \)
89 \( 1 - 4.75iT - 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−10.46021533806274292805681390740, −9.939207501947572389351345058267, −8.719857101197772572365844889828, −7.87308629934598267513513257584, −7.46313995355028965952508621787, −6.43843388374315474725821702184, −5.63431124935279209359917254450, −4.32557231889215055249189064834, −3.38349055922607800158665461392, −2.80177009647278137886383118218, 0.24806589507998957627831852386, 2.05042035287656365967656520647, 3.19321783704133084831310979431, 3.99123465014948890510206967138, 4.92182176725203059315358556741, 5.96155917717426019798940727006, 7.11308457755859246173571850043, 8.224603390237631692448901326543, 8.948710545450627241663122543451, 9.834114345486565224814601764388

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