Properties

Label 2-3744-104.51-c0-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.4035600763\)
\(L(\frac12)\) \(\approx\) \(0.4035600763\)
\(L(1)\) 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

−8.728481323333251503511201870831, −8.158696444600926723962639049710, −7.49919155086612550457125551281, −7.02103339107924165075480165321, −6.05539036938399166488887591250, −4.92944194213764110226708443291, −4.48851162588584534685624536520, −3.55064850449872361463417715247, −2.80244367298260897693151654602, −1.44957650514385746260008170959, 0.23638912551924799575035268591, 1.75662254459451993643855768097, 3.22032935971700129934475445052, 3.61567961242528486957376533627, 4.53243875330064760256872348327, 5.29869567702182160911535182821, 6.40666850339013434706927517219, 6.87775997071742131338237171996, 7.916457056309603293895373732166, 8.259272526010509065886764315779

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