Properties

Label 2-3744-13.5-c0-0-1
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: $\chi_{3744} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.471 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.192848549\)
\(L(\frac12)\) \(\approx\) \(1.192848549\)
\(L(1)\) 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

−9.057067246200666537592514096457, −7.946214172053046832023024798469, −7.18832892158794456330727502187, −6.91526742812058316580299619704, −5.68449123990573229757842408095, −5.14886062211918036381687984094, −4.15364199325891527044518540868, −3.49622933300726983214470310497, −2.28640986458408499638749451288, −1.41616799565102257221151359798, 0.71509770487713379626285831726, 2.13919396080004741856435479982, 3.00672376694689320482926807469, 3.99837260937074428732086605241, 4.71357490351977842195358177600, 5.59771683929236974383654218971, 6.54564021487880075869404897288, 6.83549714583265655702384124310, 8.010412743061647922127680967610, 8.537063889734051958640116225418

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