Properties

Label 2-4032-12.11-c1-0-5
Degree $2$
Conductor $4032$
Sign $0.577 - 0.816i$
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  − 3.62i·5-s + i·7-s − 5.03·11-s − 3.12·13-s + 6.45i·17-s − 4i·19-s + 5.03·23-s − 8.12·25-s + 8.65i·29-s − 10.2i·31-s + 3.62·35-s + 2.87·37-s + 9.27i·41-s − 4i·43-s + 1.24·47-s + ⋯
L(s)  = 1  − 1.62i·5-s + 0.377i·7-s − 1.51·11-s − 0.866·13-s + 1.56i·17-s − 0.917i·19-s + 1.05·23-s − 1.62·25-s + 1.60i·29-s − 1.84i·31-s + 0.612·35-s + 0.472·37-s + 1.44i·41-s − 0.609i·43-s + 0.180·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(32.1956\)
Root analytic conductor: \(5.67412\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4032} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4032,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9185197556\)
\(L(\frac12)\) \(\approx\) \(0.9185197556\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good5 \( 1 + 3.62iT - 5T^{2} \)
11 \( 1 + 5.03T + 11T^{2} \)
13 \( 1 + 3.12T + 13T^{2} \)
17 \( 1 - 6.45iT - 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 - 5.03T + 23T^{2} \)
29 \( 1 - 8.65iT - 29T^{2} \)
31 \( 1 + 10.2iT - 31T^{2} \)
37 \( 1 - 2.87T + 37T^{2} \)
41 \( 1 - 9.27iT - 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 1.24T + 47T^{2} \)
53 \( 1 - 8.65iT - 53T^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 3.12iT - 67T^{2} \)
71 \( 1 - 9.45T + 71T^{2} \)
73 \( 1 + 7.12T + 73T^{2} \)
79 \( 1 - 13.3iT - 79T^{2} \)
83 \( 1 - 8.83T + 83T^{2} \)
89 \( 1 - 3.62iT - 89T^{2} \)
97 \( 1 + 7.12T + 97T^{2} \)
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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

−8.522014343291637319801123511647, −7.976671646859328428919153770664, −7.30209545906806559245237409976, −6.14657840333416010421508325034, −5.37175749284015583534410534447, −4.91381851229428597551712523174, −4.24828765105218327297070135919, −2.96722129731254649902363476665, −2.09076391659608180863808912719, −0.944261313856705002575691191638, 0.30246188345683392244262985063, 2.15471515085529776824861625493, 2.85548694917096672746643027579, 3.38927207561104897307754180923, 4.66906898474850277400855299987, 5.30791968115667971720714777434, 6.24504803578800753107782458203, 7.08086930211710344318907006622, 7.44468718443960558389409312512, 8.021615798282678604514596940227

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