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Properties

Label 2-4032-48.11-c1-0-13
Degree $2$
Conductor $4032$
Sign $0.0121 - 0.999i$
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  + (0.495 − 0.495i)5-s + 7-s + (0.675 + 0.675i)11-s + (−2.39 + 2.39i)13-s + 1.03i·17-s + (−0.913 − 0.913i)19-s + 2.92i·23-s + 4.50i·25-s + (3.39 + 3.39i)29-s − 9.43i·31-s + (0.495 − 0.495i)35-s + (5.47 + 5.47i)37-s − 6.91·41-s + (2.30 − 2.30i)43-s − 9.68·47-s + ⋯
L(s)  = 1  + (0.221 − 0.221i)5-s + 0.377·7-s + (0.203 + 0.203i)11-s + (−0.664 + 0.664i)13-s + 0.251i·17-s + (−0.209 − 0.209i)19-s + 0.610i·23-s + 0.901i·25-s + (0.629 + 0.629i)29-s − 1.69i·31-s + (0.0836 − 0.0836i)35-s + (0.900 + 0.900i)37-s − 1.07·41-s + (0.351 − 0.351i)43-s − 1.41·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0121 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0121 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.0121 - 0.999i$
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: \((2,\ 4032,\ (\ :1/2),\ 0.0121 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.487794432\)
\(L(\frac12)\) \(\approx\) \(1.487794432\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.495 + 0.495i)T - 5iT^{2} \)
11 \( 1 + (-0.675 - 0.675i)T + 11iT^{2} \)
13 \( 1 + (2.39 - 2.39i)T - 13iT^{2} \)
17 \( 1 - 1.03iT - 17T^{2} \)
19 \( 1 + (0.913 + 0.913i)T + 19iT^{2} \)
23 \( 1 - 2.92iT - 23T^{2} \)
29 \( 1 + (-3.39 - 3.39i)T + 29iT^{2} \)
31 \( 1 + 9.43iT - 31T^{2} \)
37 \( 1 + (-5.47 - 5.47i)T + 37iT^{2} \)
41 \( 1 + 6.91T + 41T^{2} \)
43 \( 1 + (-2.30 + 2.30i)T - 43iT^{2} \)
47 \( 1 + 9.68T + 47T^{2} \)
53 \( 1 + (8.02 - 8.02i)T - 53iT^{2} \)
59 \( 1 + (-1.14 - 1.14i)T + 59iT^{2} \)
61 \( 1 + (7.05 - 7.05i)T - 61iT^{2} \)
67 \( 1 + (-5.66 - 5.66i)T + 67iT^{2} \)
71 \( 1 - 7.26iT - 71T^{2} \)
73 \( 1 - 6.66iT - 73T^{2} \)
79 \( 1 + 2.26iT - 79T^{2} \)
83 \( 1 + (-0.619 + 0.619i)T - 83iT^{2} \)
89 \( 1 - 3.99T + 89T^{2} \)
97 \( 1 - 6.82T + 97T^{2} \)
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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

−8.642871582499076869530016588879, −7.87063579216792850230418301771, −7.19787967296241359139145191963, −6.44433090942522106462219109313, −5.62986739253861402794293757624, −4.80666438812941016207454826984, −4.22104085893665700247833973327, −3.14591701503913069928439002036, −2.11782978477849145438896101583, −1.25465378154817080522002245388, 0.42876523454140052411812463330, 1.77597653088096640565019223439, 2.71872621676647095597390065495, 3.52104304102749144593405142741, 4.69370605718422668149165392531, 5.08801579207850505833922962847, 6.24967546624151994659677026883, 6.61158232453580396232510029804, 7.69517752060462587246204829227, 8.159259542372966775603453228649

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