Properties

Label 2-3744-13.8-c0-0-0
Degree $2$
Conductor $3744$
Sign $-0.471 - 0.881i$
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 + i)11-s − 13-s + 2i·23-s i·25-s + (−1 + i)37-s + (−1 + i)47-s + i·49-s + (−1 + i)59-s + (−1 − i)71-s + (1 − i)73-s + (1 + i)83-s + (1 + i)97-s + 2·107-s + (−1 − i)109-s + ⋯
L(s)  = 1  + (−1 + i)11-s − 13-s + 2i·23-s i·25-s + (−1 + i)37-s + (−1 + i)47-s + i·49-s + (−1 + i)59-s + (−1 − i)71-s + (1 − i)73-s + (1 + i)83-s + (1 + i)97-s + 2·107-s + (−1 − i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.471 - 0.881i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (2881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ -0.471 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6813400115\)
\(L(\frac12)\) \(\approx\) \(0.6813400115\)
\(L(1)\) 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 \( 1 + iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - 2iT - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (1 - i)T - iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (1 - i)T - iT^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + (1 + i)T + iT^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-1 - i)T + iT^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (-1 - i)T + iT^{2} \)
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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

−9.060915216984344029401056407430, −7.85884678801776605512345357631, −7.67715184209909167718325525339, −6.82294545154454196974248832912, −5.92636738845667807504939595264, −5.00830964200502202475707942690, −4.61373105437148096088441184588, −3.40484264466651294328229669767, −2.54752532615215223820573898486, −1.61027865595398589909353528693, 0.36287447309263931697076082015, 2.03471703486190963481334392981, 2.86597792640677678563461773135, 3.71101086828011157979238802297, 4.85934667144354491399694721526, 5.31066003577159365900450029141, 6.24097908528318041254709554502, 7.01610783570122534131349390971, 7.74654126851050169345744502390, 8.479114707396776627801895371896

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