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Properties

Label 2-3724-532.235-c0-0-0
Degree $2$
Conductor $3724$
Sign $0.819 + 0.572i$
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  + (−0.866 − 0.5i)2-s − i·3-s + (0.499 + 0.866i)4-s + (−0.5 + 0.866i)6-s − 0.999i·8-s + (0.866 + 0.5i)11-s + (0.866 − 0.499i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + 17-s + (−0.866 − 0.5i)19-s + (−0.499 − 0.866i)22-s + i·23-s − 0.999·24-s + (0.5 + 0.866i)25-s + (0.866 − 0.499i)26-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s − i·3-s + (0.499 + 0.866i)4-s + (−0.5 + 0.866i)6-s − 0.999i·8-s + (0.866 + 0.5i)11-s + (0.866 − 0.499i)12-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + 17-s + (−0.866 − 0.5i)19-s + (−0.499 − 0.866i)22-s + i·23-s − 0.999·24-s + (0.5 + 0.866i)25-s + (0.866 − 0.499i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.819 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.819 + 0.572i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (1831, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ 0.819 + 0.572i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9181332113\)
\(L(\frac12)\) \(\approx\) \(0.9181332113\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 \)
19 \( 1 + (0.866 + 0.5i)T \)
good3 \( 1 + iT - T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - T + T^{2} \)
23 \( 1 - iT - T^{2} \)
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 \( 1 - iT - T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 - iT - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T + T^{2} \)
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

−8.783456539366638247634897692873, −7.70645607654163311886571795050, −7.28881571760920164365942065988, −6.78679642912267598782257699815, −5.98451050244762089054383753541, −4.66607619563106021995891614038, −3.82267299498096356478842059310, −2.78471561205651034663565027839, −1.76997680021518263871389220766, −1.21527501937594374734942990013, 0.828455816192545428733805836470, 2.23407706485340615820324520272, 3.33991873070882306674740091296, 4.31189483679209421745145827693, 5.03397882619168056456695700032, 5.96943811696745743380534299014, 6.46466031063644413771005280836, 7.48187280003242573649514083040, 8.161879328427434682921540902917, 8.775582841141948357161768394155

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