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Properties

Label 2-420-21.5-c1-0-3
Degree $2$
Conductor $420$
Sign $0.221 - 0.975i$
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  + (0.979 + 1.42i)3-s + (0.5 − 0.866i)5-s + (−0.456 + 2.60i)7-s + (−1.08 + 2.79i)9-s + (0.698 − 0.403i)11-s + 3.86i·13-s + (1.72 − 0.133i)15-s + (−1.05 − 1.83i)17-s + (2.70 + 1.56i)19-s + (−4.17 + 1.89i)21-s + (4.21 + 2.43i)23-s + (−0.499 − 0.866i)25-s + (−5.05 + 1.19i)27-s − 6.67i·29-s + (5.65 − 3.26i)31-s + ⋯
L(s)  = 1  + (0.565 + 0.824i)3-s + (0.223 − 0.387i)5-s + (−0.172 + 0.985i)7-s + (−0.360 + 0.932i)9-s + (0.210 − 0.121i)11-s + 1.07i·13-s + (0.445 − 0.0344i)15-s + (−0.257 − 0.445i)17-s + (0.620 + 0.358i)19-s + (−0.910 + 0.414i)21-s + (0.879 + 0.508i)23-s + (−0.0999 − 0.173i)25-s + (−0.973 + 0.229i)27-s − 1.23i·29-s + (1.01 − 0.585i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.221 - 0.975i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.221 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28347 + 1.02506i\)
\(L(\frac12)\) \(\approx\) \(1.28347 + 1.02506i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.979 - 1.42i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.456 - 2.60i)T \)
good11 \( 1 + (-0.698 + 0.403i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.86iT - 13T^{2} \)
17 \( 1 + (1.05 + 1.83i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.70 - 1.56i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.21 - 2.43i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.67iT - 29T^{2} \)
31 \( 1 + (-5.65 + 3.26i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.89 - 6.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.44T + 41T^{2} \)
43 \( 1 + 0.819T + 43T^{2} \)
47 \( 1 + (-1.38 + 2.40i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-11.5 + 6.67i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.86 - 4.95i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.79 - 1.03i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.45 + 9.44i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (-2.67 + 1.54i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.76 + 11.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 + (1.60 - 2.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.01iT - 97T^{2} \)
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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

−11.60448973234052319728466402759, −10.22032831192090655719715919959, −9.437112775516869985154349433767, −8.896898087523798231578149442304, −8.018192296766050149531068384507, −6.62478322993740337091047277874, −5.44911154489771275392548715267, −4.59101891341523113889907124714, −3.32108269682448875489956636855, −2.08499251404113868543980640584, 1.08997491998934139769388866792, 2.75513251753761082689450940639, 3.72814948360219399748619766156, 5.31510949330984381351577988257, 6.67761106085128371043298566613, 7.13971561172921312548617509523, 8.166794002533890842598183231089, 9.069641591404454024303865657192, 10.20670258835900518473699073819, 10.85706735944883731798412842549

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