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Properties

Label 2-4032-28.27-c1-0-12
Degree $2$
Conductor $4032$
Sign $-i$
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  − 2.64i·7-s + 5.29i·11-s + 5.29i·23-s + 5·25-s − 2·29-s − 6·37-s − 5.29i·43-s − 7.00·49-s − 10·53-s + 15.8i·67-s + 5.29i·71-s + 14.0·77-s + 15.8i·79-s + 5.29i·107-s + 18·109-s + ⋯
L(s)  = 1  − 0.999i·7-s + 1.59i·11-s + 1.10i·23-s + 25-s − 0.371·29-s − 0.986·37-s − 0.806i·43-s − 49-s − 1.37·53-s + 1.93i·67-s + 0.627i·71-s + 1.59·77-s + 1.78i·79-s + 0.511i·107-s + 1.72·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $-i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (3583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.283810032\)
\(L(\frac12)\) \(\approx\) \(1.283810032\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64iT \)
good5 \( 1 - 5T^{2} \)
11 \( 1 - 5.29iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 5.29iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 5.29iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 15.8iT - 67T^{2} \)
71 \( 1 - 5.29iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 15.8iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.609574947517546075962265079641, −7.70904374231593601463597270838, −7.13279832378630315776845523664, −6.72757210428033329034350247874, −5.54077655308713678843217833674, −4.80396695372781443002574279281, −4.10490703329801993545648018676, −3.29792091092080667278049004127, −2.10629265148899205655608464761, −1.19128981666386702921935889814, 0.38360277681249166375749790802, 1.74118179592128553306189160726, 2.88198313905676551098254086385, 3.36588401064525294584337415258, 4.61390478063896448842881982765, 5.30617935676239931266497629420, 6.14612933323413287775395457128, 6.52701038884928792590406135849, 7.68810055061322622935243639120, 8.411715803373460716983072648287

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