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Properties

Label 2-3736-3736.331-c0-0-0
Degree $2$
Conductor $3736$
Sign $0.972 - 0.234i$
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  + (−0.934 − 0.356i)2-s + (0.761 + 0.375i)3-s + (0.746 + 0.665i)4-s + (−0.577 − 0.621i)6-s + (−0.460 − 0.887i)8-s + (−0.170 − 0.222i)9-s + (0.780 + 1.41i)11-s + (0.318 + 0.786i)12-s + (0.114 + 0.993i)16-s + (1.14 − 0.333i)17-s + (0.0799 + 0.268i)18-s + (0.154 − 0.0816i)19-s + (−0.227 − 1.59i)22-s + (−0.0171 − 0.848i)24-s + (0.00674 − 0.999i)25-s + ⋯
L(s)  = 1  + (−0.934 − 0.356i)2-s + (0.761 + 0.375i)3-s + (0.746 + 0.665i)4-s + (−0.577 − 0.621i)6-s + (−0.460 − 0.887i)8-s + (−0.170 − 0.222i)9-s + (0.780 + 1.41i)11-s + (0.318 + 0.786i)12-s + (0.114 + 0.993i)16-s + (1.14 − 0.333i)17-s + (0.0799 + 0.268i)18-s + (0.154 − 0.0816i)19-s + (−0.227 − 1.59i)22-s + (−0.0171 − 0.848i)24-s + (0.00674 − 0.999i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3736\)    =    \(2^{3} \cdot 467\)
Sign: $0.972 - 0.234i$
Analytic conductor: \(1.86450\)
Root analytic conductor: \(1.36546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3736} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3736,\ (\ :0),\ 0.972 - 0.234i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.169124547\)
\(L(\frac12)\) \(\approx\) \(1.169124547\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.934 + 0.356i)T \)
467 \( 1 + (-0.996 - 0.0808i)T \)
good3 \( 1 + (-0.761 - 0.375i)T + (0.608 + 0.793i)T^{2} \)
5 \( 1 + (-0.00674 + 0.999i)T^{2} \)
7 \( 1 + (-0.586 - 0.809i)T^{2} \)
11 \( 1 + (-0.780 - 1.41i)T + (-0.530 + 0.847i)T^{2} \)
13 \( 1 + (0.989 - 0.147i)T^{2} \)
17 \( 1 + (-1.14 + 0.333i)T + (0.843 - 0.536i)T^{2} \)
19 \( 1 + (-0.154 + 0.0816i)T + (0.564 - 0.825i)T^{2} \)
23 \( 1 + (-0.496 - 0.868i)T^{2} \)
29 \( 1 + (0.412 + 0.911i)T^{2} \)
31 \( 1 + (-0.194 - 0.980i)T^{2} \)
37 \( 1 + (-0.246 + 0.969i)T^{2} \)
41 \( 1 + (-1.80 - 0.742i)T + (0.709 + 0.704i)T^{2} \)
43 \( 1 + (1.77 - 0.439i)T + (0.884 - 0.466i)T^{2} \)
47 \( 1 + (0.181 - 0.983i)T^{2} \)
53 \( 1 + (0.890 + 0.454i)T^{2} \)
59 \( 1 + (-0.412 - 1.31i)T + (-0.821 + 0.570i)T^{2} \)
61 \( 1 + (-0.948 + 0.317i)T^{2} \)
67 \( 1 + (0.0937 - 0.0101i)T + (0.976 - 0.214i)T^{2} \)
71 \( 1 + (-0.650 + 0.759i)T^{2} \)
73 \( 1 + (0.262 + 1.02i)T + (-0.878 + 0.478i)T^{2} \)
79 \( 1 + (-0.764 + 0.645i)T^{2} \)
83 \( 1 + (-0.489 + 0.0663i)T + (0.963 - 0.266i)T^{2} \)
89 \( 1 + (-0.859 + 1.15i)T + (-0.285 - 0.958i)T^{2} \)
97 \( 1 + (0.0954 - 0.177i)T + (-0.553 - 0.832i)T^{2} \)
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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

−9.014621333701364450286719733630, −8.005919109854730669013337879297, −7.58616297827145745875598353725, −6.70756398487340456705362681590, −5.99782740915386978208655637063, −4.64593073501733617684608536029, −3.89846793511669952545444565321, −3.05009821626356655063249324457, −2.29143716812973581453616428857, −1.16924488958893977236247728051, 1.02632505496397135130671105229, 1.97623188511137197086979039546, 3.06339991123886471378266044783, 3.68084467972310925882450861233, 5.30990786260566424474827130278, 5.76577643623119935858272995001, 6.67640429098516471939022113061, 7.40348312874109900299159390448, 8.097676005654333657125961072164, 8.544516448318536779236427832889

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