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Properties

Label 2-4020-67.37-c1-0-19
Degree $2$
Conductor $4020$
Sign $0.921 - 0.387i$
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 + (−1.59 + 2.76i)7-s + 9-s + (2.11 − 3.66i)11-s + (1.09 + 1.88i)13-s − 15-s + (−2.56 − 4.43i)17-s + (−0.325 − 0.564i)19-s + (1.59 − 2.76i)21-s + (4.02 + 6.96i)23-s + 25-s − 27-s + (−3.94 + 6.83i)29-s + (4.23 − 7.34i)31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + (−0.602 + 1.04i)7-s + 0.333·9-s + (0.638 − 1.10i)11-s + (0.302 + 0.523i)13-s − 0.258·15-s + (−0.621 − 1.07i)17-s + (−0.0747 − 0.129i)19-s + (0.347 − 0.602i)21-s + (0.838 + 1.45i)23-s + 0.200·25-s − 0.192·27-s + (−0.733 + 1.27i)29-s + (0.761 − 1.31i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.921 - 0.387i$
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: \((2,\ 4020,\ (\ :1/2),\ 0.921 - 0.387i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.562949375\)
\(L(\frac12)\) \(\approx\) \(1.562949375\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.59 + 5.97i)T \)
good7 \( 1 + (1.59 - 2.76i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.11 + 3.66i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.09 - 1.88i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.56 + 4.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.325 + 0.564i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.02 - 6.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.94 - 6.83i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.23 + 7.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.75 + 3.03i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.135 + 0.234i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 8.32T + 43T^{2} \)
47 \( 1 + (0.171 - 0.297i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 7.89T + 53T^{2} \)
59 \( 1 + 4.39T + 59T^{2} \)
61 \( 1 + (7.05 + 12.2i)T + (-30.5 + 52.8i)T^{2} \)
71 \( 1 + (-1.45 + 2.51i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.18 - 5.51i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.598 + 1.03i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.963 - 1.66i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.23T + 89T^{2} \)
97 \( 1 + (-5.58 - 9.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.834918720449767846210195869349, −7.65278292555379184987498264851, −6.81155515879638511274564564770, −6.20445648894534011015953716717, −5.62290629488595472155805944514, −4.97941165329859241402556608251, −3.82657633013203640532752956288, −3.01067192267472527827193574935, −2.03226374180278064634066167084, −0.798963982856257673985654999283, 0.70308317341682448871715088178, 1.73594354325011122156613726747, 2.89313945188613768049542772971, 4.12915228874347625224774600124, 4.38526242637266340904440570214, 5.51550609275166861628065317202, 6.38207965980780403112369416568, 6.75716799671788973692770937819, 7.45509242838139544429537272385, 8.448993553663534082547952844653

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