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Properties

Label 2-4020-67.29-c1-0-36
Degree $2$
Conductor $4020$
Sign $-0.936 + 0.349i$
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 + (−2.18 − 3.78i)7-s + 9-s + (−0.774 − 1.34i)11-s + (2.02 − 3.50i)13-s + 15-s + (1.99 − 3.45i)17-s + (2.00 − 3.48i)19-s + (2.18 + 3.78i)21-s + (2.40 − 4.16i)23-s + 25-s − 27-s + (0.387 + 0.670i)29-s + (−0.951 − 1.64i)31-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + (−0.825 − 1.42i)7-s + 0.333·9-s + (−0.233 − 0.404i)11-s + (0.561 − 0.972i)13-s + 0.258·15-s + (0.483 − 0.837i)17-s + (0.461 − 0.798i)19-s + (0.476 + 0.825i)21-s + (0.501 − 0.868i)23-s + 0.200·25-s − 0.192·27-s + (0.0719 + 0.124i)29-s + (−0.170 − 0.296i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.936 + 0.349i$
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: \((2,\ 4020,\ (\ :1/2),\ -0.936 + 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.051587559\)
\(L(\frac12)\) \(\approx\) \(1.051587559\)
\(L(\frac{3}{2})\) 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 \( 1 + (-8.16 - 0.514i)T \)
good7 \( 1 + (2.18 + 3.78i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.774 + 1.34i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.02 + 3.50i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.99 + 3.45i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.00 + 3.48i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.40 + 4.16i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.387 - 0.670i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.951 + 1.64i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.51 + 4.36i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.44 - 4.22i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 3.63T + 43T^{2} \)
47 \( 1 + (-3.40 - 5.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.93T + 53T^{2} \)
59 \( 1 + 9.02T + 59T^{2} \)
61 \( 1 + (-2.44 + 4.24i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (5.16 + 8.94i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-1.95 + 3.39i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.84 + 10.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.02 + 8.71i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 11.9T + 89T^{2} \)
97 \( 1 + (-0.0931 + 0.161i)T + (-48.5 - 84.0i)T^{2} \)
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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

−7.72641238013173354451936782089, −7.53445417479470741996793934686, −6.63911910793771316073307198633, −6.01658131206665973858322973394, −5.06553381734859914256363202598, −4.33274550362671971850734686810, −3.44444650484822241735854499253, −2.83416396355676067335333633893, −0.908984262733137172382576946522, −0.45506657709271855771121219957, 1.33513471245450864958631504789, 2.40436419094410910866723680547, 3.45285820159762980688789251264, 4.13637085525497146668062592877, 5.29819837345143029961134462940, 5.75560724404439339598052219641, 6.47820003976668774540648439385, 7.19315681323730985437100035592, 8.088276958153393874374117181076, 8.813630280509425976571008266486

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