Properties

Label 2-420-60.59-c1-0-40
Degree $2$
Conductor $420$
Sign $0.993 + 0.111i$
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 + (2.44 − 0.140i)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 + (1.88 − 2.90i)12-s + 1.89i·13-s + (0.835 − 1.14i)14-s + (−1.56 + 3.54i)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 + (0.998 − 0.0574i)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 + (0.543 − 0.839i)12-s + 0.524i·13-s + (0.223 − 0.304i)14-s + (−0.405 + 0.914i)15-s + (−0.818 + 0.574i)16-s + 0.276·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.993 + 0.111i$
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: \((2,\ 420,\ (\ :1/2),\ 0.993 + 0.111i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.37537 - 0.132337i\)
\(L(\frac12)\) \(\approx\) \(2.37537 - 0.132337i\)
\(L(\frac{3}{2})\) 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

−11.20822423743695479333261995172, −10.38310199823424405902618039675, −9.469456981305286299950115294100, −8.981225597799270489321245219642, −7.41910979193041177654448983771, −6.25418128538615455074961187735, −5.07436612849118233598200416153, −4.08744286755495355888268657025, −3.03597880506167268154064148991, −2.01413902754149378612222467916, 1.54898627994113520815872536121, 3.23749000233078838930528540860, 4.43315845006297842323589090967, 5.66325777951156073206922414393, 6.40091357015312444951906726912, 7.61103738340364389575417072897, 8.309390418075724492866591321169, 8.955216836775243830040107474914, 10.04110852281352022947695730873, 11.73661535659481206275624809490

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