Properties

Label 2-3744-39.17-c0-0-3
Degree $2$
Conductor $3744$
Sign $0.996 + 0.0789i$
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  + 1.93·5-s + (−0.866 − 0.5i)13-s + (−0.448 + 0.258i)17-s + 2.73·25-s + (0.448 + 0.258i)29-s + (1.5 + 0.866i)37-s + (0.965 − 1.67i)41-s + (−0.5 + 0.866i)49-s − 1.93i·53-s + (0.5 + 0.866i)61-s + (−1.67 − 0.965i)65-s + i·73-s + (−0.866 + 0.499i)85-s + (−0.707 + 1.22i)89-s + (−1.73 + i)97-s + ⋯
L(s)  = 1  + 1.93·5-s + (−0.866 − 0.5i)13-s + (−0.448 + 0.258i)17-s + 2.73·25-s + (0.448 + 0.258i)29-s + (1.5 + 0.866i)37-s + (0.965 − 1.67i)41-s + (−0.5 + 0.866i)49-s − 1.93i·53-s + (0.5 + 0.866i)61-s + (−1.67 − 0.965i)65-s + i·73-s + (−0.866 + 0.499i)85-s + (−0.707 + 1.22i)89-s + (−1.73 + i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.996 + 0.0789i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (1889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.996 + 0.0789i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.806648305\)
\(L(\frac12)\) \(\approx\) \(1.806648305\)
\(L(1)\) 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 \( 1 \)
13 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 - 1.93T + T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 1.93iT - T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.809212841233249902455450660237, −8.035862612576995870794850751152, −6.98638313657643453060730594710, −6.44951242272000288221353924297, −5.61522890454844786618288097460, −5.17604773334090368313626116067, −4.20503723473854909030115212129, −2.80668106459176900027088376373, −2.32514868442386745283247908404, −1.24764769047300592284468345400, 1.31166574340507824067166883914, 2.32708077872887672256588313137, 2.80845420613287839305376662671, 4.33870088417111174022506727127, 4.98383701691080539538033742122, 5.83277893691023702348938830909, 6.36059362370148001559094736779, 7.06178741779140955030656016325, 7.977059817645799617135060391026, 8.971952137476307500949770380907

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