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Properties

Label 2-4032-48.35-c1-0-9
Degree $2$
Conductor $4032$
Sign $0.995 - 0.0912i$
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  + (−2.51 − 2.51i)5-s − 7-s + (0.984 − 0.984i)11-s + (−3.26 − 3.26i)13-s + 5.50i·17-s + (1.21 − 1.21i)19-s + 8.62i·23-s + 7.65i·25-s + (−2.04 + 2.04i)29-s − 0.164i·31-s + (2.51 + 2.51i)35-s + (−8.31 + 8.31i)37-s + 9.56·41-s + (−5.01 − 5.01i)43-s − 3.37·47-s + ⋯
L(s)  = 1  + (−1.12 − 1.12i)5-s − 0.377·7-s + (0.296 − 0.296i)11-s + (−0.904 − 0.904i)13-s + 1.33i·17-s + (0.278 − 0.278i)19-s + 1.79i·23-s + 1.53i·25-s + (−0.380 + 0.380i)29-s − 0.0295i·31-s + (0.425 + 0.425i)35-s + (−1.36 + 1.36i)37-s + 1.49·41-s + (−0.765 − 0.765i)43-s − 0.492·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.995 - 0.0912i$
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: \((2,\ 4032,\ (\ :1/2),\ 0.995 - 0.0912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9595091146\)
\(L(\frac12)\) \(\approx\) \(0.9595091146\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.51 + 2.51i)T + 5iT^{2} \)
11 \( 1 + (-0.984 + 0.984i)T - 11iT^{2} \)
13 \( 1 + (3.26 + 3.26i)T + 13iT^{2} \)
17 \( 1 - 5.50iT - 17T^{2} \)
19 \( 1 + (-1.21 + 1.21i)T - 19iT^{2} \)
23 \( 1 - 8.62iT - 23T^{2} \)
29 \( 1 + (2.04 - 2.04i)T - 29iT^{2} \)
31 \( 1 + 0.164iT - 31T^{2} \)
37 \( 1 + (8.31 - 8.31i)T - 37iT^{2} \)
41 \( 1 - 9.56T + 41T^{2} \)
43 \( 1 + (5.01 + 5.01i)T + 43iT^{2} \)
47 \( 1 + 3.37T + 47T^{2} \)
53 \( 1 + (-3.72 - 3.72i)T + 53iT^{2} \)
59 \( 1 + (-10.0 + 10.0i)T - 59iT^{2} \)
61 \( 1 + (1.07 + 1.07i)T + 61iT^{2} \)
67 \( 1 + (-3.12 + 3.12i)T - 67iT^{2} \)
71 \( 1 + 7.66iT - 71T^{2} \)
73 \( 1 + 8.40iT - 73T^{2} \)
79 \( 1 + 13.8iT - 79T^{2} \)
83 \( 1 + (-7.31 - 7.31i)T + 83iT^{2} \)
89 \( 1 - 7.49T + 89T^{2} \)
97 \( 1 + 7.84T + 97T^{2} \)
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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

−8.393867705502385091922240554633, −7.78628220561142225443211625797, −7.23185004587352536945146766982, −6.17474620044298629220170267025, −5.29595151495772096913341098825, −4.79094319026546946612136800699, −3.65941025516154919889701480642, −3.37579107299227699935773614497, −1.80041340095874601332272479623, −0.68323412823179568740150700108, 0.43831887739740624210913634665, 2.28166548333722834532368320405, 2.87557658648741989242745502437, 3.90371860490421914531526873977, 4.41678427571817977146032824010, 5.43056735185383425382643006482, 6.59610150513786353268855322194, 7.01446687494802954327283001192, 7.44435111263935172160276064137, 8.349760778751844500000435501823

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