Properties

Label 2-4020-201.200-c1-0-42
Degree $2$
Conductor $4020$
Sign $0.764 - 0.644i$
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  + (1.68 − 0.417i)3-s − 5-s + 2.42i·7-s + (2.65 − 1.40i)9-s + 2.52·11-s + 0.420i·13-s + (−1.68 + 0.417i)15-s + 1.53i·17-s − 1.99·19-s + (1.01 + 4.08i)21-s + 2.90i·23-s + 25-s + (3.87 − 3.46i)27-s + 1.32i·29-s − 4.77i·31-s + ⋯
L(s)  = 1  + (0.970 − 0.240i)3-s − 0.447·5-s + 0.917i·7-s + (0.883 − 0.467i)9-s + 0.762·11-s + 0.116i·13-s + (−0.434 + 0.107i)15-s + 0.371i·17-s − 0.456·19-s + (0.221 + 0.890i)21-s + 0.606i·23-s + 0.200·25-s + (0.745 − 0.666i)27-s + 0.245i·29-s − 0.856i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.764 - 0.644i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ 0.764 - 0.644i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.646393507\)
\(L(\frac12)\) \(\approx\) \(2.646393507\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.68 + 0.417i)T \)
5 \( 1 + T \)
67 \( 1 + (-7.34 + 3.61i)T \)
good7 \( 1 - 2.42iT - 7T^{2} \)
11 \( 1 - 2.52T + 11T^{2} \)
13 \( 1 - 0.420iT - 13T^{2} \)
17 \( 1 - 1.53iT - 17T^{2} \)
19 \( 1 + 1.99T + 19T^{2} \)
23 \( 1 - 2.90iT - 23T^{2} \)
29 \( 1 - 1.32iT - 29T^{2} \)
31 \( 1 + 4.77iT - 31T^{2} \)
37 \( 1 - 9.45T + 37T^{2} \)
41 \( 1 + 2.00T + 41T^{2} \)
43 \( 1 - 2.63iT - 43T^{2} \)
47 \( 1 - 2.60iT - 47T^{2} \)
53 \( 1 + 1.37T + 53T^{2} \)
59 \( 1 - 14.2iT - 59T^{2} \)
61 \( 1 - 11.2iT - 61T^{2} \)
71 \( 1 - 3.60iT - 71T^{2} \)
73 \( 1 + 1.92T + 73T^{2} \)
79 \( 1 - 7.44iT - 79T^{2} \)
83 \( 1 + 13.4iT - 83T^{2} \)
89 \( 1 + 2.46iT - 89T^{2} \)
97 \( 1 - 4.51iT - 97T^{2} \)
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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

−8.572312932428567873844775392493, −7.87634093011151687552743330390, −7.22482009972085211878313785539, −6.36603010211350567596505478228, −5.71491862712358306930924017869, −4.47182289289130548756741965246, −3.92378164858048611056336303473, −2.97029975150945564233094154748, −2.20761539874600550994216250489, −1.17115960304260652645019318615, 0.74560994039348872467318853073, 1.92987328122089662877498856229, 3.00294206345096525799139552864, 3.79821669846159529703485310198, 4.33022976522561696984971702890, 5.10943604623212498701542223100, 6.48940528512651828788905264581, 6.94045803008036794551779300870, 7.80597477828599615988849764304, 8.259633174329648935726085724585

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