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Properties

Label 2-4020-67.29-c1-0-13
Degree $2$
Conductor $4020$
Sign $0.155 - 0.987i$
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 + (1.80 + 3.12i)7-s + 9-s + (−2.06 − 3.58i)11-s + (−0.00353 + 0.00612i)13-s + 15-s + (0.119 − 0.207i)17-s + (−2.51 + 4.35i)19-s + (1.80 + 3.12i)21-s + (−4.10 + 7.10i)23-s + 25-s + 27-s + (2.96 + 5.13i)29-s + (−1.87 − 3.25i)31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + (0.682 + 1.18i)7-s + 0.333·9-s + (−0.623 − 1.08i)11-s + (−0.000980 + 0.00169i)13-s + 0.258·15-s + (0.0290 − 0.0503i)17-s + (−0.577 + 0.999i)19-s + (0.393 + 0.682i)21-s + (−0.855 + 1.48i)23-s + 0.200·25-s + 0.192·27-s + (0.550 + 0.953i)29-s + (−0.337 − 0.584i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
Sign: $0.155 - 0.987i$
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: \((2,\ 4020,\ (\ :1/2),\ 0.155 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.371717755\)
\(L(\frac12)\) \(\approx\) \(2.371717755\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.46 - 6.86i)T \)
good7 \( 1 + (-1.80 - 3.12i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.06 + 3.58i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.00353 - 0.00612i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.119 + 0.207i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.51 - 4.35i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.10 - 7.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.96 - 5.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.87 + 3.25i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.44 + 2.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.72 - 8.18i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 6.49T + 43T^{2} \)
47 \( 1 + (-4.62 - 8.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.25T + 53T^{2} \)
59 \( 1 - 6.12T + 59T^{2} \)
61 \( 1 + (-1.34 + 2.32i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-4.22 - 7.32i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.07 - 10.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.69 + 6.39i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.71 + 4.69i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 14.0T + 89T^{2} \)
97 \( 1 + (-2.17 + 3.76i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.506345881701833714807168755905, −8.098345446523196731708521654707, −7.36800505963648164147145224491, −6.09707735407882588552742619671, −5.75501900906950563059448786400, −5.00594480735981083687179680674, −3.92089356456171802558215204207, −3.01173425911829205365026949218, −2.24812307850452183776636560546, −1.38492486658510261269379794500, 0.60386369856910273438940147108, 1.96246454381206580792120741640, 2.48366814689897901271080746284, 3.78369430363784575004930080871, 4.53457375160959157591720725177, 4.98175456938394444488009752564, 6.23298575473994629446193409922, 6.99117268863059098040258878274, 7.52891461958744740764609783064, 8.267833623239374354435963687223

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