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Properties

Label 2-4020-67.29-c1-0-37
Degree $2$
Conductor $4020$
Sign $-0.230 + 0.973i$
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 + (0.134 + 0.232i)7-s + 9-s + (−3.05 − 5.29i)11-s + (−1.14 + 1.98i)13-s + 15-s + (−3.76 + 6.52i)17-s + (−1.88 + 3.25i)19-s + (0.134 + 0.232i)21-s + (2.57 − 4.46i)23-s + 25-s + 27-s + (−3.06 − 5.31i)29-s + (−5.25 − 9.10i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + (0.0508 + 0.0880i)7-s + 0.333·9-s + (−0.922 − 1.59i)11-s + (−0.318 + 0.551i)13-s + 0.258·15-s + (−0.913 + 1.58i)17-s + (−0.431 + 0.747i)19-s + (0.0293 + 0.0508i)21-s + (0.537 − 0.930i)23-s + 0.200·25-s + 0.192·27-s + (−0.569 − 0.986i)29-s + (−0.943 − 1.63i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.230 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $-0.230 + 0.973i$
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: \((2,\ 4020,\ (\ :1/2),\ -0.230 + 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.552151232\)
\(L(\frac12)\) \(\approx\) \(1.552151232\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.96 + 6.50i)T \)
good7 \( 1 + (-0.134 - 0.232i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.05 + 5.29i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.14 - 1.98i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.76 - 6.52i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.88 - 3.25i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.57 + 4.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.06 + 5.31i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.25 + 9.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.62 + 4.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.80 - 3.13i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 + (6.55 + 11.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12.0T + 53T^{2} \)
59 \( 1 + 11.8T + 59T^{2} \)
61 \( 1 + (-4.89 + 8.47i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (6.02 + 10.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.167 + 0.289i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.90 - 8.49i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.16 + 7.20i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 1.47T + 89T^{2} \)
97 \( 1 + (2.64 - 4.58i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.172067728219022309598328888122, −7.82657985532152754722055442724, −6.63879604428936337908576861033, −6.01733490515086848629581628287, −5.42149852381632939747908541976, −4.20301224022801294758569924507, −3.68631652240236644835967839206, −2.45697432412665481345968948169, −2.00464830629251197438883492662, −0.38222384403601907449044159830, 1.34664004128115626744197958444, 2.52784902342162146583024003636, 2.82296766639936053283594948039, 4.21297139807059537796582464403, 4.96411205402492127990445271691, 5.41017532035786842826046063137, 6.76682957459342973903174492589, 7.31353749185264329144988692102, 7.65212367629257321966608905503, 9.002429486007286838739021117755

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