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Properties

Label 2-420-140.139-c1-0-31
Degree $2$
Conductor $420$
Sign $0.939 + 0.342i$
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  + (1.36 + 0.356i)2-s − i·3-s + (1.74 + 0.976i)4-s + (−1.65 − 1.50i)5-s + (0.356 − 1.36i)6-s + (2.07 + 1.64i)7-s + (2.03 + 1.95i)8-s − 9-s + (−1.72 − 2.65i)10-s − 4.97i·11-s + (0.976 − 1.74i)12-s + 5.35·13-s + (2.25 + 2.98i)14-s + (−1.50 + 1.65i)15-s + (2.09 + 3.40i)16-s + 3.34·17-s + ⋯
L(s)  = 1  + (0.967 + 0.252i)2-s − 0.577i·3-s + (0.872 + 0.488i)4-s + (−0.738 − 0.673i)5-s + (0.145 − 0.558i)6-s + (0.783 + 0.621i)7-s + (0.721 + 0.692i)8-s − 0.333·9-s + (−0.544 − 0.838i)10-s − 1.49i·11-s + (0.282 − 0.503i)12-s + 1.48·13-s + (0.601 + 0.798i)14-s + (−0.388 + 0.426i)15-s + (0.522 + 0.852i)16-s + 0.810·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(420\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(3.35371\)
Root analytic conductor: \(1.83131\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{420} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 420,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.39386 - 0.422307i\)
\(L(\frac12)\) \(\approx\) \(2.39386 - 0.422307i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.356i)T \)
3 \( 1 + iT \)
5 \( 1 + (1.65 + 1.50i)T \)
7 \( 1 + (-2.07 - 1.64i)T \)
good11 \( 1 + 4.97iT - 11T^{2} \)
13 \( 1 - 5.35T + 13T^{2} \)
17 \( 1 - 3.34T + 17T^{2} \)
19 \( 1 + 2.12T + 19T^{2} \)
23 \( 1 + 6.71T + 23T^{2} \)
29 \( 1 + 1.42T + 29T^{2} \)
31 \( 1 + 5.76T + 31T^{2} \)
37 \( 1 + 0.864iT - 37T^{2} \)
41 \( 1 + 1.68iT - 41T^{2} \)
43 \( 1 + 5.29T + 43T^{2} \)
47 \( 1 - 8.84iT - 47T^{2} \)
53 \( 1 - 10.1iT - 53T^{2} \)
59 \( 1 - 4.51T + 59T^{2} \)
61 \( 1 - 3.54iT - 61T^{2} \)
67 \( 1 + 0.310T + 67T^{2} \)
71 \( 1 - 4.74iT - 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 + 9.38iT - 79T^{2} \)
83 \( 1 + 6.05iT - 83T^{2} \)
89 \( 1 - 3.30iT - 89T^{2} \)
97 \( 1 + 5.32T + 97T^{2} \)
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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

−11.41739607179834537805877008290, −10.83799175581887366882061026208, −8.748129844981852513134347088705, −8.297922666060738865128105234282, −7.52946236487177006744600353891, −5.99320744288814134986456920920, −5.64669499529936576170098040503, −4.21946431398531367371999530793, −3.24380523043590792118196974501, −1.50829304025938536627829137672, 1.90208516415405285855456577899, 3.63203001898718256584101496291, 4.10493888724063172695236003405, 5.20181028304608369764060100563, 6.46369513345088988840256317679, 7.41032890776208585527674338468, 8.276133156410522377178273454623, 9.966272689200275005596056494137, 10.49574086172680161743674775295, 11.34544022296546210036035376602

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