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Properties

Label 2-3736-3736.299-c0-0-0
Degree $2$
Conductor $3736$
Sign $-0.999 - 0.0150i$
Analytic cond. $1.86450$
Root an. cond. $1.36546$
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  + (−0.789 + 0.613i)2-s + (0.784 + 1.73i)3-s + (0.246 − 0.969i)4-s + (−1.68 − 0.888i)6-s + (0.399 + 0.916i)8-s + (−1.73 + 1.97i)9-s + (0.921 + 1.77i)11-s + (1.87 − 0.332i)12-s + (−0.878 − 0.478i)16-s + (0.947 + 1.10i)17-s + (0.159 − 2.62i)18-s + (0.590 − 1.58i)19-s + (−1.81 − 0.836i)22-s + (−1.27 + 1.41i)24-s + (−0.718 + 0.695i)25-s + ⋯
L(s)  = 1  + (−0.789 + 0.613i)2-s + (0.784 + 1.73i)3-s + (0.246 − 0.969i)4-s + (−1.68 − 0.888i)6-s + (0.399 + 0.916i)8-s + (−1.73 + 1.97i)9-s + (0.921 + 1.77i)11-s + (1.87 − 0.332i)12-s + (−0.878 − 0.478i)16-s + (0.947 + 1.10i)17-s + (0.159 − 2.62i)18-s + (0.590 − 1.58i)19-s + (−1.81 − 0.836i)22-s + (−1.27 + 1.41i)24-s + (−0.718 + 0.695i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3736\)    =    \(2^{3} \cdot 467\)
Sign: $-0.999 - 0.0150i$
Analytic conductor: \(1.86450\)
Root analytic conductor: \(1.36546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3736} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3736,\ (\ :0),\ -0.999 - 0.0150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.185659305\)
\(L(\frac12)\) \(\approx\) \(1.185659305\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.789 - 0.613i)T \)
467 \( 1 + (0.979 + 0.200i)T \)
good3 \( 1 + (-0.784 - 1.73i)T + (-0.660 + 0.750i)T^{2} \)
5 \( 1 + (0.718 - 0.695i)T^{2} \)
7 \( 1 + (-0.709 + 0.704i)T^{2} \)
11 \( 1 + (-0.921 - 1.77i)T + (-0.575 + 0.817i)T^{2} \)
13 \( 1 + (0.362 + 0.932i)T^{2} \)
17 \( 1 + (-0.947 - 1.10i)T + (-0.154 + 0.988i)T^{2} \)
19 \( 1 + (-0.590 + 1.58i)T + (-0.755 - 0.655i)T^{2} \)
23 \( 1 + (-0.871 + 0.490i)T^{2} \)
29 \( 1 + (-0.272 - 0.962i)T^{2} \)
31 \( 1 + (-0.956 - 0.292i)T^{2} \)
37 \( 1 + (-0.986 + 0.161i)T^{2} \)
41 \( 1 + (-0.899 + 0.606i)T + (0.374 - 0.927i)T^{2} \)
43 \( 1 + (-1.42 + 0.986i)T + (0.349 - 0.936i)T^{2} \)
47 \( 1 + (-0.324 - 0.945i)T^{2} \)
53 \( 1 + (0.924 - 0.381i)T^{2} \)
59 \( 1 + (0.872 + 0.0235i)T + (0.998 + 0.0539i)T^{2} \)
61 \( 1 + (-0.690 + 0.723i)T^{2} \)
67 \( 1 + (1.19 - 0.329i)T + (0.858 - 0.513i)T^{2} \)
71 \( 1 + (0.553 + 0.832i)T^{2} \)
73 \( 1 + (1.13 + 0.185i)T + (0.948 + 0.317i)T^{2} \)
79 \( 1 + (0.181 + 0.983i)T^{2} \)
83 \( 1 + (1.86 - 0.652i)T + (0.781 - 0.624i)T^{2} \)
89 \( 1 + (-1.36 + 1.28i)T + (0.0606 - 0.998i)T^{2} \)
97 \( 1 + (-0.916 - 0.437i)T + (0.629 + 0.777i)T^{2} \)
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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

−9.087560361431595085471254345643, −8.680220483336525282964380637837, −7.54960353943206432384647719220, −7.27425918473405235494638845010, −5.97419396504245107804338600271, −5.25814547818176467165005355767, −4.46306789229606525020576124126, −3.90735254706109085963628405246, −2.71774975970128892421776304525, −1.73986847500091145656901917969, 0.892532576413304810987010038934, 1.41650593438865041915203649752, 2.66149635156509462425477522619, 3.18497304956938915145307502536, 3.95145170386665398396433211328, 5.94271723758951728953826057457, 6.13250934505562111891468691992, 7.23440921308135476540818382464, 7.85086671554402766740234136720, 8.113183318319255560419926972176

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