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Properties

Label 2-840-56.27-c1-0-32
Degree $2$
Conductor $840$
Sign $0.993 - 0.117i$
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  + (−0.135 + 1.40i)2-s − i·3-s + (−1.96 − 0.381i)4-s + 5-s + (1.40 + 0.135i)6-s + (−2.43 + 1.04i)7-s + (0.803 − 2.71i)8-s − 9-s + (−0.135 + 1.40i)10-s + 5.54·11-s + (−0.381 + 1.96i)12-s − 1.67·13-s + (−1.13 − 3.56i)14-s − i·15-s + (3.70 + 1.49i)16-s − 5.50i·17-s + ⋯
L(s)  = 1  + (−0.0958 + 0.995i)2-s − 0.577i·3-s + (−0.981 − 0.190i)4-s + 0.447·5-s + (0.574 + 0.0553i)6-s + (−0.918 + 0.394i)7-s + (0.284 − 0.958i)8-s − 0.333·9-s + (−0.0428 + 0.445i)10-s + 1.67·11-s + (−0.110 + 0.566i)12-s − 0.465·13-s + (−0.304 − 0.952i)14-s − 0.258i·15-s + (0.927 + 0.374i)16-s − 1.33i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.993 - 0.117i$
Analytic conductor: \(6.70743\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{840} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 840,\ (\ :1/2),\ 0.993 - 0.117i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32224 + 0.0777568i\)
\(L(\frac12)\) \(\approx\) \(1.32224 + 0.0777568i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.135 - 1.40i)T \)
3 \( 1 + iT \)
5 \( 1 - T \)
7 \( 1 + (2.43 - 1.04i)T \)
good11 \( 1 - 5.54T + 11T^{2} \)
13 \( 1 + 1.67T + 13T^{2} \)
17 \( 1 + 5.50iT - 17T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 + 3.54iT - 23T^{2} \)
29 \( 1 - 8.46iT - 29T^{2} \)
31 \( 1 - 7.31T + 31T^{2} \)
37 \( 1 + 11.2iT - 37T^{2} \)
41 \( 1 - 0.0892iT - 41T^{2} \)
43 \( 1 - 8.68T + 43T^{2} \)
47 \( 1 - 12.7T + 47T^{2} \)
53 \( 1 - 1.26iT - 53T^{2} \)
59 \( 1 + 13.0iT - 59T^{2} \)
61 \( 1 + 2.49T + 61T^{2} \)
67 \( 1 - 7.26T + 67T^{2} \)
71 \( 1 - 6.75iT - 71T^{2} \)
73 \( 1 - 7.34iT - 73T^{2} \)
79 \( 1 - 1.15iT - 79T^{2} \)
83 \( 1 + 9.14iT - 83T^{2} \)
89 \( 1 - 4.16iT - 89T^{2} \)
97 \( 1 + 4.20iT - 97T^{2} \)
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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

−9.751786638948134823973510820803, −9.201356127108093831240426567346, −8.686848014119002327965626986813, −7.27422995487370989377274012494, −6.82547872114045725568048477605, −6.10509133388820185668183180740, −5.18900607440733807416495235023, −3.98979755609667736235846212340, −2.63494569788534737057035946276, −0.825025695732331159165250633036, 1.22249294594998003911053180693, 2.68169752316229062922431622316, 3.85326040169834135893670424297, 4.29308407756847610814314766554, 5.77709037055732098934997615563, 6.50707253414420570635711739059, 7.905964766251095436947094274030, 8.935020401669007793729776145771, 9.588516879549942885731666606787, 10.08879428231082252503900836762

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