Properties

Label 2-4020-67.29-c1-0-26
Degree $2$
Conductor $4020$
Sign $0.448 + 0.893i$
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 + (0.637 + 1.10i)7-s + 9-s + (−0.198 − 0.343i)11-s + (1.57 − 2.73i)13-s + 15-s + (−0.405 + 0.701i)17-s + (2.24 − 3.89i)19-s + (−0.637 − 1.10i)21-s + (−0.399 + 0.691i)23-s + 25-s − 27-s + (2.44 + 4.23i)29-s + (−0.134 − 0.233i)31-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + (0.241 + 0.417i)7-s + 0.333·9-s + (−0.0597 − 0.103i)11-s + (0.437 − 0.757i)13-s + 0.258·15-s + (−0.0982 + 0.170i)17-s + (0.515 − 0.892i)19-s + (−0.139 − 0.241i)21-s + (−0.0832 + 0.144i)23-s + 0.200·25-s − 0.192·27-s + (0.454 + 0.787i)29-s + (−0.0242 − 0.0419i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.448 + 0.893i$
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: \((2,\ 4020,\ (\ :1/2),\ 0.448 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.160878644\)
\(L(\frac12)\) \(\approx\) \(1.160878644\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.366 + 8.17i)T \)
good7 \( 1 + (-0.637 - 1.10i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.198 + 0.343i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.57 + 2.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.405 - 0.701i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.24 + 3.89i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.399 - 0.691i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.44 - 4.23i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.134 + 0.233i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.34 - 4.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.47 + 6.01i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 2.23T + 43T^{2} \)
47 \( 1 + (-1.85 - 3.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 13.4T + 53T^{2} \)
59 \( 1 - 4.24T + 59T^{2} \)
61 \( 1 + (1.93 - 3.34i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-2.11 - 3.66i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.97 + 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.255 + 0.442i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.92 + 3.34i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 0.151T + 89T^{2} \)
97 \( 1 + (-3.70 + 6.41i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.321184264662958831380184924048, −7.57718686881061823115386782069, −6.82276817270902074991973951434, −6.08798915012106233594360015949, −5.23199878991683042265696078592, −4.77073260213750673628241997163, −3.64545691194168901397386212390, −2.91304710361140071649302092555, −1.63892642457084189369074284504, −0.45603672213089351982205013068, 0.945907123114832266594669379893, 1.97745748178890557633293797471, 3.29605277568759525785473926079, 4.11476411781896846137870525743, 4.72987511414251221478862092179, 5.62926903962508381435732929118, 6.41835282355829746781294154539, 7.06176482320977662297657931542, 7.84026499894182873168525969254, 8.412140352764434818279958631877

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