Properties

Label 2-4032-48.35-c1-0-46
Degree $2$
Conductor $4032$
Sign $-0.947 - 0.320i$
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  + (−2.53 − 2.53i)5-s + 7-s + (−0.490 + 0.490i)11-s + (−4.21 − 4.21i)13-s − 6.71i·17-s + (5.38 − 5.38i)19-s + 1.37i·23-s + 7.88i·25-s + (1.45 − 1.45i)29-s + 2.66i·31-s + (−2.53 − 2.53i)35-s + (2.41 − 2.41i)37-s + 1.51·41-s + (3.40 + 3.40i)43-s − 13.2·47-s + ⋯
L(s)  = 1  + (−1.13 − 1.13i)5-s + 0.377·7-s + (−0.148 + 0.148i)11-s + (−1.16 − 1.16i)13-s − 1.62i·17-s + (1.23 − 1.23i)19-s + 0.285i·23-s + 1.57i·25-s + (0.270 − 0.270i)29-s + 0.477i·31-s + (−0.429 − 0.429i)35-s + (0.397 − 0.397i)37-s + 0.236·41-s + (0.519 + 0.519i)43-s − 1.93·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.947 - 0.320i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (1583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ -0.947 - 0.320i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7300367134\)
\(L(\frac12)\) \(\approx\) \(0.7300367134\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.53 + 2.53i)T + 5iT^{2} \)
11 \( 1 + (0.490 - 0.490i)T - 11iT^{2} \)
13 \( 1 + (4.21 + 4.21i)T + 13iT^{2} \)
17 \( 1 + 6.71iT - 17T^{2} \)
19 \( 1 + (-5.38 + 5.38i)T - 19iT^{2} \)
23 \( 1 - 1.37iT - 23T^{2} \)
29 \( 1 + (-1.45 + 1.45i)T - 29iT^{2} \)
31 \( 1 - 2.66iT - 31T^{2} \)
37 \( 1 + (-2.41 + 2.41i)T - 37iT^{2} \)
41 \( 1 - 1.51T + 41T^{2} \)
43 \( 1 + (-3.40 - 3.40i)T + 43iT^{2} \)
47 \( 1 + 13.2T + 47T^{2} \)
53 \( 1 + (9.42 + 9.42i)T + 53iT^{2} \)
59 \( 1 + (-5.46 + 5.46i)T - 59iT^{2} \)
61 \( 1 + (-8.16 - 8.16i)T + 61iT^{2} \)
67 \( 1 + (-2.47 + 2.47i)T - 67iT^{2} \)
71 \( 1 - 9.71iT - 71T^{2} \)
73 \( 1 + 2.38iT - 73T^{2} \)
79 \( 1 - 4.70iT - 79T^{2} \)
83 \( 1 + (7.75 + 7.75i)T + 83iT^{2} \)
89 \( 1 - 5.36T + 89T^{2} \)
97 \( 1 + 3.44T + 97T^{2} \)
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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

−7.930544341627046461606800815099, −7.50822553616885347106961779381, −6.86778440087947913879444624001, −5.29585382654761956710740640317, −5.09009754330394710231610247812, −4.48005088306598700890758427117, −3.31089287783788422079735210767, −2.58749999885336466994758373497, −1.00220512536614230387599542064, −0.25083423973538286909827191810, 1.56898438513793737538162736045, 2.62159462344124967076282176951, 3.55754469146404487466066773661, 4.12767458058036119042188345357, 4.99642538793742698699022549245, 6.08692620058908929631220473637, 6.71400571794832419528213396169, 7.55573524031156436414886606523, 7.87504742094151947042037655468, 8.641169035878734007187367249397

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