Properties

Label 2-4020-67.29-c1-0-27
Degree $2$
Conductor $4020$
Sign $0.932 + 0.361i$
Analytic cond. $32.0998$
Root an. cond. $5.66567$
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  − 3-s − 5-s + (1.82 + 3.15i)7-s + 9-s + (1.30 + 2.26i)11-s + (1.63 − 2.82i)13-s + 15-s + (2.07 − 3.59i)17-s + (0.703 − 1.21i)19-s + (−1.82 − 3.15i)21-s + (0.502 − 0.870i)23-s + 25-s − 27-s + (−1.80 − 3.12i)29-s + (−3.41 − 5.91i)31-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + (0.688 + 1.19i)7-s + 0.333·9-s + (0.393 + 0.681i)11-s + (0.452 − 0.784i)13-s + 0.258·15-s + (0.503 − 0.871i)17-s + (0.161 − 0.279i)19-s + (−0.397 − 0.688i)21-s + (0.104 − 0.181i)23-s + 0.200·25-s − 0.192·27-s + (−0.335 − 0.580i)29-s + (−0.613 − 1.06i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 + 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.932 + 0.361i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (3781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ 0.932 + 0.361i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.542702771\)
\(L(\frac12)\) \(\approx\) \(1.542702771\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
5 \( 1 + T \)
67 \( 1 + (5.76 + 5.81i)T \)
good7 \( 1 + (-1.82 - 3.15i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.30 - 2.26i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.63 + 2.82i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.07 + 3.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.703 + 1.21i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.502 + 0.870i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.80 + 3.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.41 + 5.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.80 + 4.86i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.37 - 2.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 1.05T + 43T^{2} \)
47 \( 1 + (-1.31 - 2.27i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.31T + 53T^{2} \)
59 \( 1 + 12.1T + 59T^{2} \)
61 \( 1 + (-7.36 + 12.7i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (1.79 + 3.11i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.56 + 4.43i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.86 - 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.72 - 13.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + (3.74 - 6.48i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.220649900533028723499384224619, −7.76367936655641977481661331293, −6.97515979831605243410528987739, −6.00660973812873779671148218538, −5.45198524266098246271309438475, −4.76855032408000733401888775666, −3.92044102051642504699860530218, −2.81769861839928334214072627553, −1.89880833667879276453658874642, −0.62087365996971544088658017907, 0.967290113363420507390912674048, 1.62746958558244721770788913488, 3.33287814754164887801394720329, 3.96826880465696343869451024561, 4.59600827353614749785369908814, 5.55407999600938442208491109067, 6.28743344890418733000454276019, 7.13392725908736621130839117153, 7.58709924544641874045964300808, 8.495859807927250177171259885195

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