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Properties

Label 2-3736-3736.91-c0-0-0
Degree $2$
Conductor $3736$
Sign $-0.292 + 0.956i$
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  + (−0.362 − 0.932i)2-s + (1.82 − 0.666i)3-s + (−0.737 + 0.675i)4-s + (−1.28 − 1.45i)6-s + (0.896 + 0.442i)8-s + (2.11 − 1.78i)9-s + (−1.24 + 0.594i)11-s + (−0.894 + 1.72i)12-s + (0.0875 − 0.996i)16-s + (0.0299 + 0.492i)17-s + (−2.43 − 1.32i)18-s + (1.42 − 0.878i)19-s + (1.00 + 0.946i)22-s + (1.93 + 0.209i)24-s + (−0.530 − 0.847i)25-s + ⋯
L(s)  = 1  + (−0.362 − 0.932i)2-s + (1.82 − 0.666i)3-s + (−0.737 + 0.675i)4-s + (−1.28 − 1.45i)6-s + (0.896 + 0.442i)8-s + (2.11 − 1.78i)9-s + (−1.24 + 0.594i)11-s + (−0.894 + 1.72i)12-s + (0.0875 − 0.996i)16-s + (0.0299 + 0.492i)17-s + (−2.43 − 1.32i)18-s + (1.42 − 0.878i)19-s + (1.00 + 0.946i)22-s + (1.93 + 0.209i)24-s + (−0.530 − 0.847i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.292 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3736\)    =    \(2^{3} \cdot 467\)
Sign: $-0.292 + 0.956i$
Analytic conductor: \(1.86450\)
Root analytic conductor: \(1.36546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3736} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3736,\ (\ :0),\ -0.292 + 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.884052359\)
\(L(\frac12)\) \(\approx\) \(1.884052359\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.362 + 0.932i)T \)
467 \( 1 + (-0.908 + 0.418i)T \)
good3 \( 1 + (-1.82 + 0.666i)T + (0.764 - 0.645i)T^{2} \)
5 \( 1 + (0.530 + 0.847i)T^{2} \)
7 \( 1 + (0.979 + 0.200i)T^{2} \)
11 \( 1 + (1.24 - 0.594i)T + (0.629 - 0.777i)T^{2} \)
13 \( 1 + (0.967 - 0.253i)T^{2} \)
17 \( 1 + (-0.0299 - 0.492i)T + (-0.992 + 0.121i)T^{2} \)
19 \( 1 + (-1.42 + 0.878i)T + (0.448 - 0.893i)T^{2} \)
23 \( 1 + (-0.781 - 0.624i)T^{2} \)
29 \( 1 + (0.639 - 0.768i)T^{2} \)
31 \( 1 + (-0.496 + 0.868i)T^{2} \)
37 \( 1 + (0.960 - 0.279i)T^{2} \)
41 \( 1 + (-0.999 + 1.69i)T + (-0.484 - 0.874i)T^{2} \)
43 \( 1 + (-0.540 - 1.90i)T + (-0.851 + 0.525i)T^{2} \)
47 \( 1 + (-0.564 + 0.825i)T^{2} \)
53 \( 1 + (0.100 - 0.994i)T^{2} \)
59 \( 1 + (-0.0365 + 0.773i)T + (-0.995 - 0.0942i)T^{2} \)
61 \( 1 + (0.154 - 0.988i)T^{2} \)
67 \( 1 + (-1.24 + 0.636i)T + (0.586 - 0.809i)T^{2} \)
71 \( 1 + (0.805 - 0.592i)T^{2} \)
73 \( 1 + (1.20 + 0.351i)T + (0.843 + 0.536i)T^{2} \)
79 \( 1 + (0.0740 - 0.997i)T^{2} \)
83 \( 1 + (0.373 - 1.88i)T + (-0.924 - 0.381i)T^{2} \)
89 \( 1 + (0.284 - 1.11i)T + (-0.878 - 0.478i)T^{2} \)
97 \( 1 + (-0.0115 - 1.71i)T + (-0.999 + 0.0134i)T^{2} \)
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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

−8.445872384983906284118237613935, −7.81165583781546471846222892012, −7.57278175787508374917695383651, −6.63395409231669785851728765750, −5.18953339058627505182578371803, −4.26174139005744686372017247813, −3.46408206778485508557822589292, −2.64079269305418067600610840980, −2.23124066830773589551638626294, −1.10427812700418607344909739029, 1.52044400944256469828566064749, 2.75205442061378205881902940499, 3.44651962241252775219661499857, 4.33256679494954454698685933892, 5.18486647433463787279063921273, 5.80777778057877499109098290038, 7.28979953369232551980624881731, 7.52918708407611697834535380483, 8.207888867784293850814618867218, 8.742311998289790974486326004200

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