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Properties

Label 2-4020-67.37-c1-0-3
Degree $2$
Conductor $4020$
Sign $-0.819 - 0.573i$
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 + (−2.33 + 4.05i)7-s + 9-s + (2.04 − 3.54i)11-s + (0.5 + 0.866i)13-s − 15-s + (0.405 + 0.702i)17-s + (−1.42 − 2.46i)19-s + (−2.33 + 4.05i)21-s + (2.18 + 3.78i)23-s + 25-s + 27-s + (−4.40 + 7.62i)29-s + (0.131 − 0.228i)31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + (−0.884 + 1.53i)7-s + 0.333·9-s + (0.617 − 1.07i)11-s + (0.138 + 0.240i)13-s − 0.258·15-s + (0.0983 + 0.170i)17-s + (−0.326 − 0.565i)19-s + (−0.510 + 0.884i)21-s + (0.456 + 0.790i)23-s + 0.200·25-s + 0.192·27-s + (−0.817 + 1.41i)29-s + (0.0236 − 0.0409i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.819 - 0.573i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.819 - 0.573i$
Analytic conductor: \(32.0998\)
Root analytic conductor: \(5.66567\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4020} (841, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4020,\ (\ :1/2),\ -0.819 - 0.573i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.023650200\)
\(L(\frac12)\) \(\approx\) \(1.023650200\)
\(L(\frac{3}{2})\) 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 \( 1 + (8.03 + 1.55i)T \)
good7 \( 1 + (2.33 - 4.05i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.04 + 3.54i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.405 - 0.702i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.42 + 2.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.18 - 3.78i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.40 - 7.62i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.131 + 0.228i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.572 + 0.991i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.81 - 3.15i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 0.155T + 43T^{2} \)
47 \( 1 + (1.69 - 2.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.736T + 53T^{2} \)
59 \( 1 - 3.35T + 59T^{2} \)
61 \( 1 + (-0.373 - 0.646i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-1.60 + 2.77i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.76 - 8.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.82 - 10.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.18 - 10.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 + (9.60 + 16.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.720360674609485755091408421250, −8.325572598804838177967048241603, −7.21699294349659899594391333400, −6.56222526074474013069057129737, −5.81104415706583101028287090749, −5.12238153440070535035711673221, −3.89414328194417506609692143263, −3.25486527993115173199443027836, −2.61809757908892663690688341140, −1.40666229629579026213972900674, 0.27252326291244688547156151484, 1.49001197364102746795453468155, 2.68829723742412189884675165932, 3.76941469765733882490834994366, 4.01433679684935540799167247049, 4.88471460186911593780171949312, 6.22743379747285502137805735600, 6.82612891412372180495799410655, 7.45032661243092636542677354822, 7.961056940169448329329061342897

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