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Properties

Label 2-4032-48.11-c1-0-35
Degree $2$
Conductor $4032$
Sign $-0.159 + 0.987i$
Analytic cond. $32.1956$
Root an. cond. $5.67412$
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  + (−1.18 + 1.18i)5-s + 7-s + (1.81 + 1.81i)11-s + (−3.25 + 3.25i)13-s + 1.32i·17-s + (−5.18 − 5.18i)19-s − 4.26i·23-s + 2.20i·25-s + (−4.51 − 4.51i)29-s − 5.06i·31-s + (−1.18 + 1.18i)35-s + (−3.79 − 3.79i)37-s − 0.186·41-s + (−7.38 + 7.38i)43-s + 4.80·47-s + ⋯
L(s)  = 1  + (−0.528 + 0.528i)5-s + 0.377·7-s + (0.548 + 0.548i)11-s + (−0.902 + 0.902i)13-s + 0.321i·17-s + (−1.18 − 1.18i)19-s − 0.889i·23-s + 0.441i·25-s + (−0.837 − 0.837i)29-s − 0.909i·31-s + (−0.199 + 0.199i)35-s + (−0.623 − 0.623i)37-s − 0.0291·41-s + (−1.12 + 1.12i)43-s + 0.700·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.159 + 0.987i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (3599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ -0.159 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6668081599\)
\(L(\frac12)\) \(\approx\) \(0.6668081599\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 - T \)
good5 \( 1 + (1.18 - 1.18i)T - 5iT^{2} \)
11 \( 1 + (-1.81 - 1.81i)T + 11iT^{2} \)
13 \( 1 + (3.25 - 3.25i)T - 13iT^{2} \)
17 \( 1 - 1.32iT - 17T^{2} \)
19 \( 1 + (5.18 + 5.18i)T + 19iT^{2} \)
23 \( 1 + 4.26iT - 23T^{2} \)
29 \( 1 + (4.51 + 4.51i)T + 29iT^{2} \)
31 \( 1 + 5.06iT - 31T^{2} \)
37 \( 1 + (3.79 + 3.79i)T + 37iT^{2} \)
41 \( 1 + 0.186T + 41T^{2} \)
43 \( 1 + (7.38 - 7.38i)T - 43iT^{2} \)
47 \( 1 - 4.80T + 47T^{2} \)
53 \( 1 + (-9.62 + 9.62i)T - 53iT^{2} \)
59 \( 1 + (-3.57 - 3.57i)T + 59iT^{2} \)
61 \( 1 + (-2.12 + 2.12i)T - 61iT^{2} \)
67 \( 1 + (4.70 + 4.70i)T + 67iT^{2} \)
71 \( 1 + 0.828iT - 71T^{2} \)
73 \( 1 - 7.95iT - 73T^{2} \)
79 \( 1 + 1.04iT - 79T^{2} \)
83 \( 1 + (-9.07 + 9.07i)T - 83iT^{2} \)
89 \( 1 - 10.2T + 89T^{2} \)
97 \( 1 - 17.6T + 97T^{2} \)
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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

−8.215778804864988473554964284438, −7.35768262406026107988596547025, −6.89820103347121307596054373970, −6.23284867074697210842602070209, −5.08184889018150280703069850723, −4.35171027431356009678875213305, −3.81254257017774244074581805308, −2.51318219170017241004956861922, −1.90225010553712400408439492837, −0.20388110126594839265926021898, 1.08957105225233919026288847978, 2.17876701397747973353875049454, 3.39203939904439599795022008732, 3.99769344477567371775760180503, 5.00236628600891407582001598176, 5.50306010407545713577086074692, 6.45800311736363244752914869022, 7.33969133248024600498333861177, 7.938243020221622079970785356113, 8.649610506996848557304767436677

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