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Properties

Label 2-420-105.23-c1-0-15
Degree $2$
Conductor $420$
Sign $-0.941 + 0.335i$
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  + (−0.479 − 1.66i)3-s + (−0.274 − 2.21i)5-s + (0.250 − 2.63i)7-s + (−2.53 + 1.59i)9-s + (−1.95 + 1.12i)11-s + (−1.60 − 1.60i)13-s + (−3.56 + 1.52i)15-s + (−1.10 + 4.12i)17-s + (6.82 + 3.94i)19-s + (−4.50 + 0.845i)21-s + (−2.31 − 8.63i)23-s + (−4.84 + 1.22i)25-s + (3.87 + 3.46i)27-s − 3.34·29-s + (−2.63 − 4.55i)31-s + ⋯
L(s)  = 1  + (−0.276 − 0.960i)3-s + (−0.122 − 0.992i)5-s + (0.0948 − 0.995i)7-s + (−0.846 + 0.532i)9-s + (−0.589 + 0.340i)11-s + (−0.446 − 0.446i)13-s + (−0.919 + 0.392i)15-s + (−0.268 + 1.00i)17-s + (1.56 + 0.904i)19-s + (−0.982 + 0.184i)21-s + (−0.482 − 1.80i)23-s + (−0.969 + 0.244i)25-s + (0.745 + 0.666i)27-s − 0.620·29-s + (−0.472 − 0.818i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.941 + 0.335i$
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: \((2,\ 420,\ (\ :1/2),\ -0.941 + 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.152955 - 0.884952i\)
\(L(\frac12)\) \(\approx\) \(0.152955 - 0.884952i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.479 + 1.66i)T \)
5 \( 1 + (0.274 + 2.21i)T \)
7 \( 1 + (-0.250 + 2.63i)T \)
good11 \( 1 + (1.95 - 1.12i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.60 + 1.60i)T + 13iT^{2} \)
17 \( 1 + (1.10 - 4.12i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-6.82 - 3.94i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.31 + 8.63i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 3.34T + 29T^{2} \)
31 \( 1 + (2.63 + 4.55i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.379 + 1.41i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.07iT - 41T^{2} \)
43 \( 1 + (4.16 + 4.16i)T + 43iT^{2} \)
47 \( 1 + (-12.1 + 3.26i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-4.15 - 1.11i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.733 + 1.26i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.02 - 6.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.8 - 2.91i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 9.99iT - 71T^{2} \)
73 \( 1 + (-0.353 + 1.31i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.66 - 3.26i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.33 + 7.33i)T - 83iT^{2} \)
89 \( 1 + (-4.63 + 8.02i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.06 + 7.06i)T - 97iT^{2} \)
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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

−10.77394519157985021678949891165, −10.06108702767750981099044040843, −8.733175376737628496183996316445, −7.82201569480898737830679257514, −7.33374944416291520890820216669, −5.97435109738662578215713830036, −5.09144345548229048684394782188, −3.87042641946805520880705447205, −2.02898994861099715084435360246, −0.58365001489056705960027888814, 2.62273814302084834338507523335, 3.46623434002518555176191417598, 5.03216605086729532457955895516, 5.64585914114560483484729964751, 6.91013344267002615960494712873, 7.904461031075868748084789109334, 9.353294737552115965275925674614, 9.562949413767805803891891686250, 10.84490137084099444499735959889, 11.51508373195233289739730009538

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