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Properties

Label 2-420-105.89-c1-0-8
Degree $2$
Conductor $420$
Sign $0.623 - 0.781i$
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  + (1.48 + 0.884i)3-s + (2.23 − 0.0584i)5-s + (−0.0973 + 2.64i)7-s + (1.43 + 2.63i)9-s + (−2.62 + 1.51i)11-s − 2.31·13-s + (3.38 + 1.89i)15-s + (4.49 − 2.59i)17-s + (−5.58 − 3.22i)19-s + (−2.48 + 3.85i)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  + (0.859 + 0.510i)3-s + (0.999 − 0.0261i)5-s + (−0.0368 + 0.999i)7-s + (0.478 + 0.878i)9-s + (−0.792 + 0.457i)11-s − 0.642·13-s + (0.872 + 0.488i)15-s + (1.09 − 0.629i)17-s + (−1.28 − 0.739i)19-s + (−0.542 + 0.840i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.623 - 0.781i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.623 - 0.781i$
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: \((2,\ 420,\ (\ :1/2),\ 0.623 - 0.781i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81534 + 0.873874i\)
\(L(\frac12)\) \(\approx\) \(1.81534 + 0.873874i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.48 - 0.884i)T \)
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

−11.11711828610818752785365161870, −10.08986574055636031125594714958, −9.594559758656768444735672875313, −8.739323819817929499985208566176, −7.86584559027416600914310180914, −6.62208511726272174470195688917, −5.36341622246105187322838073247, −4.65348961644800546959860742322, −2.81859934189664951135630688528, −2.24575970030482598345518597007, 1.39611824156106768638160627390, 2.72726430377474248751324451531, 3.90018675537364059106157175501, 5.40584068909821564044172399884, 6.47724940671295241139830204984, 7.46315310740951637979806203615, 8.215899618129314865096905968341, 9.286133681405262746500289955382, 10.17957711707701037055756092749, 10.70883249805387113171277505175

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