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Properties

Label 2-3744-8.5-c1-0-29
Degree $2$
Conductor $3744$
Sign $0.707 + 0.707i$
Analytic cond. $29.8959$
Root an. cond. $5.46772$
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  − 3i·5-s − 3·7-s + i·13-s + 7·17-s + 4i·19-s + 4·23-s − 4·25-s + 4i·29-s + 8·31-s + 9i·35-s + 7i·37-s − 2·41-s + i·43-s − 7·47-s + 2·49-s + ⋯
L(s)  = 1  − 1.34i·5-s − 1.13·7-s + 0.277i·13-s + 1.69·17-s + 0.917i·19-s + 0.834·23-s − 0.800·25-s + 0.742i·29-s + 1.43·31-s + 1.52i·35-s + 1.15i·37-s − 0.312·41-s + 0.152i·43-s − 1.02·47-s + 0.285·49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(29.8959\)
Root analytic conductor: \(5.46772\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (1873, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.662086679\)
\(L(\frac12)\) \(\approx\) \(1.662086679\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good5 \( 1 + 3iT - 5T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 - 7T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 4iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 7iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 + 7T + 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 + 14iT - 59T^{2} \)
61 \( 1 + 10iT - 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−8.249013359938070707737154342917, −8.024390805779863929082350778109, −6.78827086121344587444916827575, −6.24822188770034146979853787686, −5.24572452337337432413576212737, −4.83755644608983131410532151537, −3.63313328700375840805774317030, −3.11027091550498953658062136111, −1.61379954473299415405957426706, −0.72156226730330465486695067466, 0.814596216461762226005109417355, 2.50667611440250328149732812498, 3.06833981906607753311022033292, 3.66409819739302987864936745574, 4.85360871593317142099655212774, 5.89642221190825359384461806898, 6.36091404248984226508671355346, 7.18313079836657899187790598726, 7.58448174844116243181029221418, 8.634062321997018886648688134846

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