Properties

Label 2-3724-532.11-c0-0-0
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: $\chi_{3724} (3203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ 0.983 + 0.182i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4392295639\)
\(L(\frac12)\) \(\approx\) \(0.4392295639\)
\(L(1)\) 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

−8.452835346745081115083108489485, −7.925133043039206103061765521945, −7.08765356738598637752699838309, −6.36620104429970610590609122435, −5.66336607753320709288782514045, −5.11123936771592119183489251876, −4.31015622373403251912049788081, −3.65369288460554169556934239256, −2.11525661783365455683824574787, −0.31312699152461624148476796347, 1.19116476975732416251713516843, 2.02035688785779943015252161692, 3.21979818444840351160258814261, 4.08278202384591875999618649238, 4.79714441016903614037997862214, 5.78589952608206587564666818171, 6.36528851443485811562798285468, 7.10644488464723404205566703346, 8.187981378069398338861974893220, 8.842697116992846785043981322275

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