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Properties

Label 2-420-105.23-c1-0-7
Degree $2$
Conductor $420$
Sign $0.677 - 0.735i$
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  + (1.62 + 0.591i)3-s + (1.94 + 1.10i)5-s + (−2.63 + 0.225i)7-s + (2.30 + 1.92i)9-s + (−1.86 + 1.07i)11-s + (3.89 + 3.89i)13-s + (2.51 + 2.94i)15-s + (1.09 − 4.10i)17-s + (0.631 + 0.364i)19-s + (−4.42 − 1.19i)21-s + (−1.48 − 5.55i)23-s + (2.57 + 4.28i)25-s + (2.60 + 4.49i)27-s − 3.95·29-s + (−2.33 − 4.04i)31-s + ⋯
L(s)  = 1  + (0.939 + 0.341i)3-s + (0.870 + 0.492i)5-s + (−0.996 + 0.0853i)7-s + (0.767 + 0.641i)9-s + (−0.561 + 0.324i)11-s + (1.08 + 1.08i)13-s + (0.650 + 0.759i)15-s + (0.266 − 0.994i)17-s + (0.144 + 0.0835i)19-s + (−0.965 − 0.259i)21-s + (−0.310 − 1.15i)23-s + (0.515 + 0.857i)25-s + (0.502 + 0.864i)27-s − 0.735·29-s + (−0.418 − 0.725i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.677 - 0.735i$
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: \((2,\ 420,\ (\ :1/2),\ 0.677 - 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80666 + 0.792272i\)
\(L(\frac12)\) \(\approx\) \(1.80666 + 0.792272i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 0.591i)T \)
5 \( 1 + (-1.94 - 1.10i)T \)
7 \( 1 + (2.63 - 0.225i)T \)
good11 \( 1 + (1.86 - 1.07i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.89 - 3.89i)T + 13iT^{2} \)
17 \( 1 + (-1.09 + 4.10i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.631 - 0.364i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.48 + 5.55i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 3.95T + 29T^{2} \)
31 \( 1 + (2.33 + 4.04i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.598 + 2.23i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.95iT - 41T^{2} \)
43 \( 1 + (-3.57 - 3.57i)T + 43iT^{2} \)
47 \( 1 + (-9.34 + 2.50i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.22 + 1.66i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (4.01 + 6.94i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.20 - 10.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.606 + 0.162i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 4.00iT - 71T^{2} \)
73 \( 1 + (-2.79 + 10.4i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (6.61 + 3.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.56 - 6.56i)T - 83iT^{2} \)
89 \( 1 + (-0.656 + 1.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.0 + 10.0i)T - 97iT^{2} \)
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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

−11.00871285404396698765440631734, −10.24716938377168375505499729323, −9.385340545849060022558091707080, −8.967878919481817075022046483745, −7.56732757606127976921702089840, −6.69634017214577690712194629922, −5.67725365187951451324751680504, −4.22763983935026692356037921508, −3.07999033644322950200124264182, −2.08808607986920091668542143215, 1.35684548422457450397872916914, 2.89020028111275008731861348006, 3.79307671081264256280037741142, 5.57423325511127865307503255992, 6.24173104382732435688313550926, 7.53241572007471144817148806716, 8.429750589800766414742318774873, 9.214291005726946799129626500009, 10.02573392650054554597715990825, 10.79044357993964397996365835682

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