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Properties

Label 2-3744-39.35-c0-0-3
Degree $2$
Conductor $3744$
Sign $-0.644 + 0.764i$
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  − 1.93i·5-s + (−0.866 − 0.5i)13-s + (1.67 − 0.965i)17-s − 2.73·25-s + (−1.67 − 0.965i)29-s + (−0.5 + 0.866i)37-s + (1.67 + 0.965i)41-s + (0.5 − 0.866i)49-s − 0.517i·53-s + (0.5 + 0.866i)61-s + (−0.965 + 1.67i)65-s − 1.73·73-s + (−1.86 − 3.23i)85-s + (−1.22 − 0.707i)89-s + (−0.448 − 0.258i)101-s + ⋯
L(s)  = 1  − 1.93i·5-s + (−0.866 − 0.5i)13-s + (1.67 − 0.965i)17-s − 2.73·25-s + (−1.67 − 0.965i)29-s + (−0.5 + 0.866i)37-s + (1.67 + 0.965i)41-s + (0.5 − 0.866i)49-s − 0.517i·53-s + (0.5 + 0.866i)61-s + (−0.965 + 1.67i)65-s − 1.73·73-s + (−1.86 − 3.23i)85-s + (−1.22 − 0.707i)89-s + (−0.448 − 0.258i)101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3744\)    =    \(2^{5} \cdot 3^{2} \cdot 13\)
Sign: $-0.644 + 0.764i$
Analytic conductor: \(1.86849\)
Root analytic conductor: \(1.36693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3744} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3744,\ (\ :0),\ -0.644 + 0.764i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.076462154\)
\(L(\frac12)\) \(\approx\) \(1.076462154\)
\(L(1)\) 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 \( 1 + 1.93iT - T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-1.67 - 0.965i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 0.517iT - T^{2} \)
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 \( 1 + 1.73T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
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

−8.357383407638189256962530845833, −7.80385102686346594944230403554, −7.23446847490318537551860485173, −5.75023970258107132468708307182, −5.48366738267031556922294524994, −4.71224580873845358810270031856, −3.97846514671797734637377553254, −2.84820791247217315699701265109, −1.62435642125160202309411944685, −0.61364463067819719144543609907, 1.78650493047866231281346873656, 2.66536395542221955841798533751, 3.48571749892104972999427417309, 4.08900307906611114443542108425, 5.53378824582296957541214053449, 5.93497060866043359908226911230, 6.94564556253181358449412791481, 7.39334854847872025815661973460, 7.88621555127460787212236677048, 9.129691558351695660405956042045

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