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Properties

Label 2-3744-468.367-c0-0-1
Degree $2$
Conductor $3744$
Sign $0.167 - 0.985i$
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  + (0.866 + 0.5i)3-s + (0.5 + 0.866i)5-s + (0.499 + 0.866i)9-s + i·11-s + (0.5 − 0.866i)13-s + 0.999i·15-s + (−0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + 0.999i·27-s + 29-s + (0.866 − 0.5i)31-s + (−0.5 + 0.866i)33-s + (−0.5 − 0.866i)37-s + (0.866 − 0.499i)39-s + (−1.73 + i)43-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.5 + 0.866i)5-s + (0.499 + 0.866i)9-s + i·11-s + (0.5 − 0.866i)13-s + 0.999i·15-s + (−0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + 0.999i·27-s + 29-s + (0.866 − 0.5i)31-s + (−0.5 + 0.866i)33-s + (−0.5 − 0.866i)37-s + (0.866 − 0.499i)39-s + (−1.73 + i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.167 - 0.985i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (2239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.167 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.916825311\)
\(L(\frac12)\) \(\approx\) \(1.916825311\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 - iT - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T + T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + iT - T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−8.749890006019516390254637516430, −8.254415954591566187203917071974, −7.45940567693743990401905321462, −6.62141869619346775075139947034, −6.06403701712820202863178069941, −4.85079731384236906294316511365, −4.27092171469564498333660361435, −3.25330562926382989846168691734, −2.55617702249060738655183587508, −1.75927753796417887102066630359, 1.04372096855273014168819087491, 1.92768171436339025542176289774, 2.91313066662441992161650777002, 3.82502844995770270494451770519, 4.68330555851764144650490720359, 5.55085081617080478907232300568, 6.57640802195147812402176566992, 6.84847675359935633429081502409, 8.121128714803134101401139188413, 8.665177626192124777039541792062

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