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Properties

Label 2-3744-104.51-c0-0-5
Degree $2$
Conductor $3744$
Sign $0.707 + 0.707i$
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  + 1.41·5-s − 1.41i·11-s − i·13-s + 1.00·25-s − 1.41i·41-s − 2·43-s + 1.41·47-s − 49-s − 2.00i·55-s − 1.41i·59-s + 2i·61-s − 1.41i·65-s + 1.41·71-s + 2i·79-s + 1.41i·83-s + ⋯
L(s)  = 1  + 1.41·5-s − 1.41i·11-s − i·13-s + 1.00·25-s − 1.41i·41-s − 2·43-s + 1.41·47-s − 49-s − 2.00i·55-s − 1.41i·59-s + 2i·61-s − 1.41i·65-s + 1.41·71-s + 2i·79-s + 1.41i·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (2287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.638358815\)
\(L(\frac12)\) \(\approx\) \(1.638358815\)
\(L(1)\) 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 \( 1 + iT \)
good5 \( 1 - 1.41T + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + 2T + T^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 - 2iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 2iT - T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
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

−8.553973982884147029783967294927, −8.089712627533763209205878611376, −6.95374104216066410858638230400, −6.27375945724187630600013052961, −5.51368428833927828307701388441, −5.24639563486609219157692575452, −3.82717364803241911058310210475, −3.00040267985160145562043072486, −2.14807701760074852186929795146, −0.968231665347310459829792074464, 1.65129915570072927569738896434, 2.06051342510879132093593889764, 3.18682674729443108435621125177, 4.45505310880636672047288517141, 4.93166838064083239973352330561, 5.89825294635399597748387593535, 6.56857921515892766418432962793, 7.12244094590950342689088854813, 8.063269197268561397060553306355, 9.010266361054060627329496754757

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