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Properties

Label 2-420-140.139-c1-0-45
Degree $2$
Conductor $420$
Sign $-0.514 + 0.857i$
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  + (1.33 − 0.475i)2-s − i·3-s + (1.54 − 1.26i)4-s + (−2.22 − 0.260i)5-s + (−0.475 − 1.33i)6-s + (−1.17 − 2.36i)7-s + (1.45 − 2.42i)8-s − 9-s + (−3.08 + 0.709i)10-s − 0.365i·11-s + (−1.26 − 1.54i)12-s − 1.72·13-s + (−2.69 − 2.59i)14-s + (−0.260 + 2.22i)15-s + (0.790 − 3.92i)16-s + 0.146·17-s + ⋯
L(s)  = 1  + (0.941 − 0.336i)2-s − 0.577i·3-s + (0.773 − 0.633i)4-s + (−0.993 − 0.116i)5-s + (−0.194 − 0.543i)6-s + (−0.445 − 0.895i)7-s + (0.515 − 0.856i)8-s − 0.333·9-s + (−0.974 + 0.224i)10-s − 0.110i·11-s + (−0.365 − 0.446i)12-s − 0.479·13-s + (−0.720 − 0.693i)14-s + (−0.0672 + 0.573i)15-s + (0.197 − 0.980i)16-s + 0.0356·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.514 + 0.857i$
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: \((2,\ 420,\ (\ :1/2),\ -0.514 + 0.857i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.922421 - 1.62858i\)
\(L(\frac12)\) \(\approx\) \(0.922421 - 1.62858i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 + 0.475i)T \)
3 \( 1 + iT \)
5 \( 1 + (2.22 + 0.260i)T \)
7 \( 1 + (1.17 + 2.36i)T \)
good11 \( 1 + 0.365iT - 11T^{2} \)
13 \( 1 + 1.72T + 13T^{2} \)
17 \( 1 - 0.146T + 17T^{2} \)
19 \( 1 - 4.73T + 19T^{2} \)
23 \( 1 - 3.44T + 23T^{2} \)
29 \( 1 + 5.29T + 29T^{2} \)
31 \( 1 - 6.07T + 31T^{2} \)
37 \( 1 + 7.52iT - 37T^{2} \)
41 \( 1 - 7.16iT - 41T^{2} \)
43 \( 1 - 8.65T + 43T^{2} \)
47 \( 1 + 3.83iT - 47T^{2} \)
53 \( 1 - 11.5iT - 53T^{2} \)
59 \( 1 - 6.18T + 59T^{2} \)
61 \( 1 + 0.348iT - 61T^{2} \)
67 \( 1 + 12.1T + 67T^{2} \)
71 \( 1 - 7.78iT - 71T^{2} \)
73 \( 1 - 4.55T + 73T^{2} \)
79 \( 1 + 14.2iT - 79T^{2} \)
83 \( 1 - 3.53iT - 83T^{2} \)
89 \( 1 + 13.6iT - 89T^{2} \)
97 \( 1 - 16.1T + 97T^{2} \)
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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

−11.19390060850550223624027038619, −10.30995484281622286501514308578, −9.170088964760004491127535802587, −7.55869322355046185395369709562, −7.30520190911830932670841133557, −6.11803009141137577332668769557, −4.87577470408088475985985413247, −3.84605391871756937253010124315, −2.85924901552094051020433796513, −0.934649982183379923944415640430, 2.71344388442146061811012291091, 3.58294993761169037182781456603, 4.73058948079933120152271615269, 5.56059428724979881023003116959, 6.74572732435110147543892283810, 7.66968477117865621305988416607, 8.634862764004442876870101379818, 9.690924950468354034423648773214, 10.90411517885059283514818817336, 11.73492570979386500244505477388

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