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Properties

Label 2-3724-76.11-c0-0-0
Degree $2$
Conductor $3724$
Sign $-0.211 - 0.977i$
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 + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.499 − 0.866i)6-s + 0.999i·8-s + i·11-s − 0.999i·12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.5 + 0.866i)22-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)24-s + (0.5 + 0.866i)25-s − 0.999i·26-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.499 − 0.866i)6-s + 0.999i·8-s + i·11-s − 0.999i·12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.5 + 0.866i)22-s + (−0.866 + 0.5i)23-s + (0.5 − 0.866i)24-s + (0.5 + 0.866i)25-s − 0.999i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.211 - 0.977i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (3431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ -0.211 - 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.298706580\)
\(L(\frac12)\) \(\approx\) \(1.298706580\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 - iT - T^{2} \)
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 \( 1 - iT - T^{2} \)
37 \( 1 - T + T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−8.707714450960355549182972950797, −7.67508474291889220066424357243, −7.40302657024879329249288227642, −6.56205820749490483549265556120, −5.79617383663025969827959142817, −5.42557397654084767118418586832, −4.48528114274333365159159394804, −3.66218690047369299678616694849, −2.64118435184457573768128963360, −1.50138207965025357219596806661, 0.61543855009068511327415498179, 2.19919747748941643606839004654, 2.96686299312515203293364443793, 4.10357523413864115500672323521, 4.67342067615672037130444077548, 5.34622192644074084773659774829, 6.05183623942842235096424313033, 6.67364591255839910951324716732, 7.54746210394344365087973209816, 8.626383124553581708222495440894

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