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Properties

Label 2-3744-39.38-c0-0-3
Degree $2$
Conductor $3744$
Sign $0.577 - 0.816i$
Analytic cond. $1.86849$
Root an. cond. $1.36693$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·11-s − 13-s + 1.41i·17-s + 2i·19-s − 25-s − 1.41i·29-s + 2i·31-s + 1.41·47-s + 49-s − 1.41i·53-s − 1.41·59-s + 1.41·71-s + 1.41·83-s + 1.41i·101-s + 1.41i·113-s + ⋯
L(s)  = 1  + 1.41·11-s − 13-s + 1.41i·17-s + 2i·19-s − 25-s − 1.41i·29-s + 2i·31-s + 1.41·47-s + 49-s − 1.41i·53-s − 1.41·59-s + 1.41·71-s + 1.41·83-s + 1.41i·101-s + 1.41i·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (3041, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.577 - 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.238018449\)
\(L(\frac12)\) \(\approx\) \(1.238018449\)
\(L(1)\) 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 \( 1 \)
13 \( 1 + T \)
good5 \( 1 + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - 1.41T + T^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 - 2iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 - 2iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 + 1.41iT - T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.760774555636175791923922822930, −8.058099534161647880169786424938, −7.42340757830212746803314058924, −6.41165054577998507754911691554, −6.02057743765917445472065624291, −5.05560083039814757990789237622, −3.92122169681425779021643531621, −3.70010525925865396020261816861, −2.20691513524190528126646672082, −1.40632494557808042055630762595, 0.75204837644625229371663867573, 2.19245372625571115517203343922, 2.95335226492276597086962578365, 4.09611825033126590191091637393, 4.71369326503036973267243960281, 5.55145228863294958785239689606, 6.47893630563509988088216708841, 7.21150561800922106308495547084, 7.56942014762125695963033495797, 8.848926672720592461387452272561

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