Properties

Label 2-420-60.59-c1-0-14
Degree $2$
Conductor $420$
Sign $0.0796 - 0.996i$
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  + (−0.835 − 1.14i)2-s + (1.10 + 1.33i)3-s + (−0.602 + 1.90i)4-s + (−1 + 2i)5-s + (0.602 − 2.37i)6-s + 7-s + (2.67 − 0.907i)8-s + (−0.569 + 2.94i)9-s + (3.11 − 0.531i)10-s − 2.20·11-s + (−3.21 + 1.29i)12-s − 1.89i·13-s + (−0.835 − 1.14i)14-s + (−3.77 + 0.868i)15-s + (−3.27 − 2.29i)16-s − 1.13·17-s + ⋯
L(s)  = 1  + (−0.591 − 0.806i)2-s + (0.636 + 0.771i)3-s + (−0.301 + 0.953i)4-s + (−0.447 + 0.894i)5-s + (0.245 − 0.969i)6-s + 0.377·7-s + (0.947 − 0.320i)8-s + (−0.189 + 0.981i)9-s + (0.985 − 0.167i)10-s − 0.664·11-s + (−0.927 + 0.374i)12-s − 0.524i·13-s + (−0.223 − 0.304i)14-s + (−0.974 + 0.224i)15-s + (−0.818 − 0.574i)16-s − 0.276·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0796 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0796 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.0796 - 0.996i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.0796 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.711817 + 0.657239i\)
\(L(\frac12)\) \(\approx\) \(0.711817 + 0.657239i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.835 + 1.14i)T \)
3 \( 1 + (-1.10 - 1.33i)T \)
5 \( 1 + (1 - 2i)T \)
7 \( 1 - T \)
good11 \( 1 + 2.20T + 11T^{2} \)
13 \( 1 + 1.89iT - 13T^{2} \)
17 \( 1 + 1.13T + 17T^{2} \)
19 \( 1 - 8.56iT - 19T^{2} \)
23 \( 1 - 3.21iT - 23T^{2} \)
29 \( 1 + 1.89iT - 29T^{2} \)
31 \( 1 - 5.90iT - 31T^{2} \)
37 \( 1 - 0.409iT - 37T^{2} \)
41 \( 1 + 4.40iT - 41T^{2} \)
43 \( 1 + 0.934T + 43T^{2} \)
47 \( 1 + 2.67iT - 47T^{2} \)
53 \( 1 - 6.81T + 53T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 - 12.4T + 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 + 3.76iT - 79T^{2} \)
83 \( 1 - 6.84iT - 83T^{2} \)
89 \( 1 + 16.1iT - 89T^{2} \)
97 \( 1 + 18.4iT - 97T^{2} \)
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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

−11.16229498990974906212037669477, −10.23816234232850951173190229175, −10.07816970500568777753968936539, −8.636408556945828718324266004060, −8.031103993830072472734725825474, −7.25248724513174401802230219596, −5.43266642212026845225629409571, −4.03466984420156668370720869976, −3.28961342638728631178936007248, −2.14544638349669842420452634714, 0.70420965804479226476391564329, 2.29004087071605131451401318328, 4.30990576236128458921911936405, 5.28767148324572590132701262728, 6.61831166983850639860601132638, 7.41147692070654039884656663514, 8.264108872191281803347904761646, 8.855687301522025301282237298304, 9.601247310450138489351168565277, 11.00600383342732366283204271423

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