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Properties

Label 2-3724-532.387-c0-0-1
Degree $2$
Conductor $3724$
Sign $-0.983 + 0.182i$
Analytic cond. $1.85851$
Root an. cond. $1.36327$
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  + i·2-s + (0.866 + 0.5i)3-s − 4-s + (−0.5 + 0.866i)6-s − i·8-s + (−0.866 + 0.5i)11-s + (−0.866 − 0.5i)12-s + (−0.5 + 0.866i)13-s + 16-s + (−0.5 + 0.866i)17-s + i·19-s + (−0.5 − 0.866i)22-s + (0.866 − 0.5i)23-s + (0.5 − 0.866i)24-s − 25-s + (−0.866 − 0.5i)26-s + ⋯
L(s)  = 1  + i·2-s + (0.866 + 0.5i)3-s − 4-s + (−0.5 + 0.866i)6-s − i·8-s + (−0.866 + 0.5i)11-s + (−0.866 − 0.5i)12-s + (−0.5 + 0.866i)13-s + 16-s + (−0.5 + 0.866i)17-s + i·19-s + (−0.5 − 0.866i)22-s + (0.866 − 0.5i)23-s + (0.5 − 0.866i)24-s − 25-s + (−0.866 − 0.5i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.983 + 0.182i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (1451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ -0.983 + 0.182i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.003232461\)
\(L(\frac12)\) \(\approx\) \(1.003232461\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
7 \( 1 \)
19 \( 1 - iT \)
good3 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
11 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.816804541996119558914689176455, −8.492688176705815979486839207799, −7.58311692013344428496316959434, −7.06321783270419952731599315094, −6.13234264343387051733691414331, −5.38785929894020550555921312428, −4.45839658958472568534696101512, −3.92508099109368381308759054178, −2.98486135222564180099534865166, −1.80767061983917839512624700675, 0.48573078504062185215331963776, 2.02792009985347986408124029241, 2.64718436470699227170840401798, 3.21880676564819533748686443537, 4.28797025778055333338983187684, 5.26161375166413064905502165549, 5.74230085572199996504755303173, 7.35750218837123110875085239535, 7.59976011485268693136647452078, 8.386131189470873231627556925354

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