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Properties

Label 2-3744-13.11-c0-0-1
Degree $2$
Conductor $3744$
Sign $0.679 - 0.733i$
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  + (1.36 + 1.36i)5-s + (0.866 − 0.5i)13-s + (0.866 + 0.5i)17-s + 2.73i·25-s + (−0.866 − 1.5i)29-s + (−0.5 − 0.133i)37-s + (0.133 − 0.5i)41-s + (0.866 − 0.5i)49-s − 53-s + (−0.5 + 0.866i)61-s + (1.86 + 0.499i)65-s + (0.366 − 0.366i)73-s + (0.499 + 1.86i)85-s + (−1.36 − 0.366i)89-s + (−1.36 + 0.366i)97-s + ⋯
L(s)  = 1  + (1.36 + 1.36i)5-s + (0.866 − 0.5i)13-s + (0.866 + 0.5i)17-s + 2.73i·25-s + (−0.866 − 1.5i)29-s + (−0.5 − 0.133i)37-s + (0.133 − 0.5i)41-s + (0.866 − 0.5i)49-s − 53-s + (−0.5 + 0.866i)61-s + (1.86 + 0.499i)65-s + (0.366 − 0.366i)73-s + (0.499 + 1.86i)85-s + (−1.36 − 0.366i)89-s + (−1.36 + 0.366i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $0.679 - 0.733i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (1441, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ 0.679 - 0.733i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.757287674\)
\(L(\frac12)\) \(\approx\) \(1.757287674\)
\(L(1)\) 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 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
7 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \)
41 \( 1 + (-0.133 + 0.5i)T + (-0.866 - 0.5i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
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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

−8.885951409929734279738139365224, −7.946534578997932875425849941534, −7.24619452327737166544430137718, −6.43877007905963494698345251358, −5.84404559645065489605283250791, −5.42597543643618820134143941103, −3.96759922425607974707124128933, −3.22112120366402181385572998293, −2.38845601803049350986138644124, −1.48989542340944718525310287371, 1.18254222212804567152149447663, 1.79486288735526694084689711155, 3.01516902921276766710917105495, 4.12172388610163581162239200055, 5.01179678677169306624934971212, 5.53165402146394423312776141476, 6.19561701198576306344760105029, 7.03478967742479029198386928449, 8.080614687437988522576260074209, 8.730023862228999314535653166328

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