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Properties

Label 2-420-140.139-c1-0-36
Degree $2$
Conductor $420$
Sign $-0.501 + 0.865i$
Analytic cond. $3.35371$
Root an. cond. $1.83131$
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 + 0.903i)2-s − i·3-s + (0.366 − 1.96i)4-s + (0.660 − 2.13i)5-s + (0.903 + 1.08i)6-s + (−2.64 − 0.110i)7-s + (1.37 + 2.46i)8-s − 9-s + (1.21 + 2.92i)10-s − 2.42i·11-s + (−1.96 − 0.366i)12-s + 5.02·13-s + (2.97 − 2.26i)14-s + (−2.13 − 0.660i)15-s + (−3.73 − 1.43i)16-s − 6.95·17-s + ⋯
L(s)  = 1  + (−0.769 + 0.639i)2-s − 0.577i·3-s + (0.183 − 0.983i)4-s + (0.295 − 0.955i)5-s + (0.368 + 0.444i)6-s + (−0.999 − 0.0416i)7-s + (0.487 + 0.873i)8-s − 0.333·9-s + (0.383 + 0.923i)10-s − 0.732i·11-s + (−0.567 − 0.105i)12-s + 1.39·13-s + (0.795 − 0.606i)14-s + (−0.551 − 0.170i)15-s + (−0.932 − 0.359i)16-s − 1.68·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.501 + 0.865i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.501 + 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.501 + 0.865i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.501 + 0.865i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.304197 - 0.528013i\)
\(L(\frac12)\) \(\approx\) \(0.304197 - 0.528013i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 - 0.903i)T \)
3 \( 1 + iT \)
5 \( 1 + (-0.660 + 2.13i)T \)
7 \( 1 + (2.64 + 0.110i)T \)
good11 \( 1 + 2.42iT - 11T^{2} \)
13 \( 1 - 5.02T + 13T^{2} \)
17 \( 1 + 6.95T + 17T^{2} \)
19 \( 1 - 1.26T + 19T^{2} \)
23 \( 1 + 6.81T + 23T^{2} \)
29 \( 1 + 4.56T + 29T^{2} \)
31 \( 1 - 0.903T + 31T^{2} \)
37 \( 1 + 8.02iT - 37T^{2} \)
41 \( 1 + 3.33iT - 41T^{2} \)
43 \( 1 - 2.06T + 43T^{2} \)
47 \( 1 + 3.31iT - 47T^{2} \)
53 \( 1 + 4.77iT - 53T^{2} \)
59 \( 1 - 8.38T + 59T^{2} \)
61 \( 1 - 4.38iT - 61T^{2} \)
67 \( 1 + 8.25T + 67T^{2} \)
71 \( 1 - 11.5iT - 71T^{2} \)
73 \( 1 - 12.3T + 73T^{2} \)
79 \( 1 - 8.69iT - 79T^{2} \)
83 \( 1 + 9.72iT - 83T^{2} \)
89 \( 1 + 13.6iT - 89T^{2} \)
97 \( 1 - 1.23T + 97T^{2} \)
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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

−10.79639534139970118478454931606, −9.695240833780290926435855324079, −8.815276159281331253004614777433, −8.419107192676710820437890129651, −7.14653844372421648203368650475, −6.14201809378956938735354992481, −5.68508036553680540617340371557, −3.99617937793844082440871751181, −2.02830008914024673881135937484, −0.48047127150808931282788901839, 2.14307065538839779438263350194, 3.31562503859014014885011140956, 4.19870303186531662398909532975, 6.15210688705659305230433558515, 6.79699466369583035260578429673, 8.030594789350497126966862892996, 9.152767981123602543346794216743, 9.733928962464167562415262795698, 10.55911488683517282809747357929, 11.15239193883676004494725015124

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