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Properties

Label 2-4032-48.35-c1-0-36
Degree $2$
Conductor $4032$
Sign $-0.960 + 0.276i$
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  + (−2.96 − 2.96i)5-s − 7-s + (−0.569 + 0.569i)11-s + (2.53 + 2.53i)13-s + 1.03i·17-s + (5.23 − 5.23i)19-s − 8.71i·23-s + 12.6i·25-s + (6.05 − 6.05i)29-s − 3.00i·31-s + (2.96 + 2.96i)35-s + (−0.149 + 0.149i)37-s − 8.63·41-s + (−1.73 − 1.73i)43-s + 4.10·47-s + ⋯
L(s)  = 1  + (−1.32 − 1.32i)5-s − 0.377·7-s + (−0.171 + 0.171i)11-s + (0.703 + 0.703i)13-s + 0.251i·17-s + (1.20 − 1.20i)19-s − 1.81i·23-s + 2.52i·25-s + (1.12 − 1.12i)29-s − 0.539i·31-s + (0.501 + 0.501i)35-s + (−0.0246 + 0.0246i)37-s − 1.34·41-s + (−0.264 − 0.264i)43-s + 0.599·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.276i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.276i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-0.960 + 0.276i$
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: \((2,\ 4032,\ (\ :1/2),\ -0.960 + 0.276i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8135024106\)
\(L(\frac12)\) \(\approx\) \(0.8135024106\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.96 + 2.96i)T + 5iT^{2} \)
11 \( 1 + (0.569 - 0.569i)T - 11iT^{2} \)
13 \( 1 + (-2.53 - 2.53i)T + 13iT^{2} \)
17 \( 1 - 1.03iT - 17T^{2} \)
19 \( 1 + (-5.23 + 5.23i)T - 19iT^{2} \)
23 \( 1 + 8.71iT - 23T^{2} \)
29 \( 1 + (-6.05 + 6.05i)T - 29iT^{2} \)
31 \( 1 + 3.00iT - 31T^{2} \)
37 \( 1 + (0.149 - 0.149i)T - 37iT^{2} \)
41 \( 1 + 8.63T + 41T^{2} \)
43 \( 1 + (1.73 + 1.73i)T + 43iT^{2} \)
47 \( 1 - 4.10T + 47T^{2} \)
53 \( 1 + (-6.04 - 6.04i)T + 53iT^{2} \)
59 \( 1 + (6.05 - 6.05i)T - 59iT^{2} \)
61 \( 1 + (-5.81 - 5.81i)T + 61iT^{2} \)
67 \( 1 + (0.0256 - 0.0256i)T - 67iT^{2} \)
71 \( 1 + 14.5iT - 71T^{2} \)
73 \( 1 + 5.38iT - 73T^{2} \)
79 \( 1 + 3.89iT - 79T^{2} \)
83 \( 1 + (1.40 + 1.40i)T + 83iT^{2} \)
89 \( 1 + 17.0T + 89T^{2} \)
97 \( 1 - 3.75T + 97T^{2} \)
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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

−8.269837395721747882645131179221, −7.44888312569580585645503218335, −6.76741977923826333402850150062, −5.84998619846809768336748075359, −4.78932784648779857421847950654, −4.44041665814572809233215099539, −3.63604090032493666242230341802, −2.60456284713814329114846353344, −1.14979752466374324111892401893, −0.28940014217040018026093584116, 1.23610662840399962058062311943, 2.84042493970552985669678212154, 3.44802445301509332759414925009, 3.76871640819167478986065485329, 5.13855820668511439902613752299, 5.84335764241466487549793464199, 6.88172042469598093019765492389, 7.17638654372050024683528039604, 8.093793857346096537966372420579, 8.399454697957408693050388929171

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