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Properties

Label 2-420-105.89-c1-0-1
Degree $2$
Conductor $420$
Sign $-0.992 - 0.124i$
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  + (0.0216 + 1.73i)3-s + (−2.23 + 0.0584i)5-s + (−0.0973 + 2.64i)7-s + (−2.99 + 0.0749i)9-s + (2.62 − 1.51i)11-s − 2.31·13-s + (−0.149 − 3.87i)15-s + (−4.49 + 2.59i)17-s + (−5.58 − 3.22i)19-s + (−4.58 − 0.111i)21-s + (−2.43 + 4.21i)23-s + (4.99 − 0.261i)25-s + (−0.194 − 5.19i)27-s + 3.48i·29-s + (1.16 − 0.673i)31-s + ⋯
L(s)  = 1  + (0.0124 + 0.999i)3-s + (−0.999 + 0.0261i)5-s + (−0.0368 + 0.999i)7-s + (−0.999 + 0.0249i)9-s + (0.792 − 0.457i)11-s − 0.642·13-s + (−0.0386 − 0.999i)15-s + (−1.09 + 0.629i)17-s + (−1.28 − 0.739i)19-s + (−0.999 − 0.0243i)21-s + (−0.507 + 0.879i)23-s + (0.998 − 0.0522i)25-s + (−0.0374 − 0.999i)27-s + 0.647i·29-s + (0.209 − 0.120i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.992 - 0.124i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ -0.992 - 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0385421 + 0.616265i\)
\(L(\frac12)\) \(\approx\) \(0.0385421 + 0.616265i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.0216 - 1.73i)T \)
5 \( 1 + (2.23 - 0.0584i)T \)
7 \( 1 + (0.0973 - 2.64i)T \)
good11 \( 1 + (-2.62 + 1.51i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.31T + 13T^{2} \)
17 \( 1 + (4.49 - 2.59i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.58 + 3.22i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.43 - 4.21i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.48iT - 29T^{2} \)
31 \( 1 + (-1.16 + 0.673i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.43 - 1.40i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.06T + 41T^{2} \)
43 \( 1 - 2.42iT - 43T^{2} \)
47 \( 1 + (3.30 + 1.90i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.52 - 6.11i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.18 - 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-11.4 - 6.60i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.72 + 2.14i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.08iT - 71T^{2} \)
73 \( 1 + (-0.0763 - 0.132i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.04 - 5.27i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.2iT - 83T^{2} \)
89 \( 1 + (7.10 - 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.63T + 97T^{2} \)
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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

−11.48424847574797927739828463494, −10.91790411008943528870773891910, −9.755061244910382515544394820188, −8.743135751897744437918709861750, −8.434363741780349323085977190465, −6.90219244732675484255231212001, −5.83757130102397036804715225448, −4.64129415321547212435772451119, −3.84123509055802824982528952129, −2.57304197895965539534377122793, 0.37901673205103695586547307469, 2.17345953521610572096581160660, 3.80559906964662080999421038202, 4.69245729084474441693359080839, 6.48944118765154237092662317720, 6.96292711478152866707107038383, 7.914217775740111628735903335660, 8.630763997895359151624837140798, 9.914725765264322504304705792130, 11.02431829596721231194381303587

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