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Properties

Label 2-3744-13.12-c1-0-67
Degree $2$
Conductor $3744$
Sign $-0.554 - 0.832i$
Analytic cond. $29.8959$
Root an. cond. $5.46772$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·5-s + (−3 + 2i)13-s + 2·17-s − 11·25-s − 10·29-s + 12i·37-s − 8i·41-s + 7·49-s − 14·53-s − 10·61-s + (8 + 12i)65-s − 16i·73-s − 8i·85-s + 16i·89-s + 8i·97-s + ⋯
L(s)  = 1  − 1.78i·5-s + (−0.832 + 0.554i)13-s + 0.485·17-s − 2.20·25-s − 1.85·29-s + 1.97i·37-s − 1.24i·41-s + 49-s − 1.92·53-s − 1.28·61-s + (0.992 + 1.48i)65-s − 1.87i·73-s − 0.867i·85-s + 1.69i·89-s + 0.812i·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.554 - 0.832i$
Analytic conductor: \(29.8959\)
Root analytic conductor: \(5.46772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (3457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 3744,\ (\ :1/2),\ -0.554 - 0.832i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + (3 - 2i)T \)
good5 \( 1 + 4iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 12iT - 37T^{2} \)
41 \( 1 + 8iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 14T + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 16iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 16iT - 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−7.973832741044919213494427292927, −7.57087221163524274035714087100, −6.47969997481039407164373511739, −5.56776528716580331474882766751, −4.98250105396219012433013251598, −4.36913661305886093985718531011, −3.45790316511305003191858201870, −2.09023021001683883419116091656, −1.26214535222265802680455156179, 0, 1.87648676244566357541774297229, 2.77348086636941653821329988722, 3.38347913481718259752106087415, 4.27532854977663050671091326032, 5.50262499783384054777244257763, 6.00867871883172260252049158466, 6.94345609829164421040776940499, 7.47349600087989102088205715604, 7.920250557861090988000720899088

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