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Properties

Label 2-4020-67.29-c1-0-6
Degree $2$
Conductor $4020$
Sign $-0.522 - 0.852i$
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 + (1.18 + 2.05i)7-s + 9-s + (2.18 + 3.78i)11-s + (0.5 − 0.866i)13-s − 15-s + (−0.813 + 1.40i)17-s + (−3.18 + 5.51i)19-s + (1.18 + 2.05i)21-s + (−0.813 + 1.40i)23-s + 25-s + 27-s + (−2.18 − 3.78i)29-s + (−3.87 − 6.70i)31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + (0.448 + 0.776i)7-s + 0.333·9-s + (0.659 + 1.14i)11-s + (0.138 − 0.240i)13-s − 0.258·15-s + (−0.197 + 0.341i)17-s + (−0.730 + 1.26i)19-s + (0.258 + 0.448i)21-s + (−0.169 + 0.293i)23-s + 0.200·25-s + 0.192·27-s + (−0.405 − 0.703i)29-s + (−0.695 − 1.20i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.522 - 0.852i$
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: \((2,\ 4020,\ (\ :1/2),\ -0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.714086349\)
\(L(\frac12)\) \(\approx\) \(1.714086349\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.05 + 8.11i)T \)
good7 \( 1 + (-1.18 - 2.05i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.18 - 3.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.813 - 1.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.18 - 5.51i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.813 - 1.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.18 + 3.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.87 + 6.70i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.55 - 7.89i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.18 - 8.98i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 3.37T + 43T^{2} \)
47 \( 1 + (-0.813 - 1.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 8.74T + 53T^{2} \)
59 \( 1 + 2.74T + 59T^{2} \)
61 \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (6.55 + 11.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-6.5 + 11.2i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.12 - 3.68i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.55 - 6.16i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.4T + 89T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.623851269664806660552202448634, −7.904036712225606953059634104335, −7.54561703030185104128230336467, −6.42059135112005379161788242397, −5.86619216306706806704030333286, −4.70735929491790800629947870084, −4.16338147410362468567307596687, −3.31116731657755185330221002373, −2.17431635693623432430883163499, −1.55739000156374333286696310356, 0.44014533643502251072354119628, 1.56953958991253918438757652580, 2.73890376549569602520223501379, 3.66618168883153304172944837339, 4.19053943601754495843404879473, 5.06091092285716598876781529798, 6.06038684939817165593009062720, 7.11313645508625652654770102796, 7.24089641062311274471567813446, 8.384745820018553517931798887991

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