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Properties

Label 2-420-105.23-c1-0-14
Degree $2$
Conductor $420$
Sign $0.875 + 0.482i$
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  + (1.68 − 0.396i)3-s + (1.86 − 1.23i)5-s + (2.58 − 0.576i)7-s + (2.68 − 1.33i)9-s + (−3.73 + 2.15i)11-s + (−2 − 2i)13-s + (2.65 − 2.81i)15-s + (−1.57 + 5.89i)17-s + (−5.47 − 3.15i)19-s + (4.12 − 1.99i)21-s + (0.789 + 2.94i)23-s + (1.96 − 4.59i)25-s + (4 − 3.31i)27-s + 0.683·29-s + (3 + 5.19i)31-s + ⋯
L(s)  = 1  + (0.973 − 0.228i)3-s + (0.834 − 0.550i)5-s + (0.976 − 0.217i)7-s + (0.895 − 0.445i)9-s + (−1.12 + 0.650i)11-s + (−0.554 − 0.554i)13-s + (0.686 − 0.727i)15-s + (−0.383 + 1.43i)17-s + (−1.25 − 0.724i)19-s + (0.900 − 0.435i)21-s + (0.164 + 0.614i)23-s + (0.392 − 0.919i)25-s + (0.769 − 0.638i)27-s + 0.126·29-s + (0.538 + 0.933i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.875 + 0.482i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.875 + 0.482i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.875 + 0.482i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10138 - 0.540618i\)
\(L(\frac12)\) \(\approx\) \(2.10138 - 0.540618i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 0.396i)T \)
5 \( 1 + (-1.86 + 1.23i)T \)
7 \( 1 + (-2.58 + 0.576i)T \)
good11 \( 1 + (3.73 - 2.15i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2 + 2i)T + 13iT^{2} \)
17 \( 1 + (1.57 - 5.89i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (5.47 + 3.15i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.789 - 2.94i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 0.683T + 29T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.46 + 5.46i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 3.63iT - 41T^{2} \)
43 \( 1 + (-5.15 - 5.15i)T + 43iT^{2} \)
47 \( 1 + (6.83 - 1.83i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.933 - 0.250i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.84 + 4.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.34 - 2.32i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.24 + 1.40i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.31iT - 71T^{2} \)
73 \( 1 + (-2.42 + 9.06i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-10.6 - 6.15i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.15 - 7.15i)T - 83iT^{2} \)
89 \( 1 + (-8.29 + 14.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.68 - 2.68i)T - 97iT^{2} \)
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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

−10.79797532289126581651889684923, −10.23102799037116244979853433576, −9.189605490336424656981243557746, −8.313111168248416173399724155525, −7.72974961675300533113650483849, −6.51366129250024807735542049769, −5.14665064228276634792577350704, −4.33881758906351888663680752791, −2.58806453276402857190350818146, −1.65745719016605540606976836596, 2.09512511993075495819027813999, 2.79177017669462270308148692441, 4.43173438622015609411465282491, 5.37273208507415683913592079111, 6.70934148053538714756736507814, 7.74338266313004001759168521901, 8.546799989632606548109045237083, 9.425373690138671191775995099214, 10.36235114631924558756194420593, 10.98029694869401697582384477626

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