↑

Properties

Label 2-3736-3736.59-c0-0-0
Degree $2$
Conductor $3736$
Sign $0.653 + 0.756i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.967 − 0.253i)2-s + (0.474 − 0.673i)3-s + (0.871 + 0.490i)4-s + (−0.629 + 0.531i)6-s + (−0.718 − 0.695i)8-s + (0.108 + 0.302i)9-s + (−0.00208 − 0.308i)11-s + (0.743 − 0.354i)12-s + (0.519 + 0.854i)16-s + (1.29 + 0.705i)17-s + (−0.0280 − 0.319i)18-s + (0.352 − 1.91i)19-s + (−0.0762 + 0.299i)22-s + (−0.809 + 0.154i)24-s + (−0.553 + 0.832i)25-s + ⋯
L(s)  = 1  + (−0.967 − 0.253i)2-s + (0.474 − 0.673i)3-s + (0.871 + 0.490i)4-s + (−0.629 + 0.531i)6-s + (−0.718 − 0.695i)8-s + (0.108 + 0.302i)9-s + (−0.00208 − 0.308i)11-s + (0.743 − 0.354i)12-s + (0.519 + 0.854i)16-s + (1.29 + 0.705i)17-s + (−0.0280 − 0.319i)18-s + (0.352 − 1.91i)19-s + (−0.0762 + 0.299i)22-s + (−0.809 + 0.154i)24-s + (−0.553 + 0.832i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 + 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3736\)    =    \(2^{3} \cdot 467\)
Sign: $0.653 + 0.756i$
Analytic conductor: \(1.86450\)
Root analytic conductor: \(1.36546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3736} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3736,\ (\ :0),\ 0.653 + 0.756i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.060074821\)
\(L(\frac12)\) \(\approx\) \(1.060074821\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.967 + 0.253i)T \)
467 \( 1 + (-0.728 - 0.685i)T \)
good3 \( 1 + (-0.474 + 0.673i)T + (-0.337 - 0.941i)T^{2} \)
5 \( 1 + (0.553 - 0.832i)T^{2} \)
7 \( 1 + (-0.908 - 0.418i)T^{2} \)
11 \( 1 + (0.00208 + 0.308i)T + (-0.999 + 0.0134i)T^{2} \)
13 \( 1 + (0.943 - 0.330i)T^{2} \)
17 \( 1 + (-1.29 - 0.705i)T + (0.542 + 0.840i)T^{2} \)
19 \( 1 + (-0.352 + 1.91i)T + (-0.934 - 0.356i)T^{2} \)
23 \( 1 + (0.924 + 0.381i)T^{2} \)
29 \( 1 + (0.680 - 0.732i)T^{2} \)
31 \( 1 + (-0.963 - 0.266i)T^{2} \)
37 \( 1 + (0.285 + 0.958i)T^{2} \)
41 \( 1 + (0.790 - 0.473i)T + (0.472 - 0.881i)T^{2} \)
43 \( 1 + (-0.299 - 0.359i)T + (-0.181 + 0.983i)T^{2} \)
47 \( 1 + (0.127 - 0.991i)T^{2} \)
53 \( 1 + (-0.976 - 0.214i)T^{2} \)
59 \( 1 + (0.568 + 0.182i)T + (0.813 + 0.581i)T^{2} \)
61 \( 1 + (0.992 + 0.121i)T^{2} \)
67 \( 1 + (-1.64 - 0.0664i)T + (0.996 + 0.0808i)T^{2} \)
71 \( 1 + (-0.948 - 0.317i)T^{2} \)
73 \( 1 + (-0.571 + 1.91i)T + (-0.836 - 0.547i)T^{2} \)
79 \( 1 + (0.259 - 0.965i)T^{2} \)
83 \( 1 + (0.383 + 0.424i)T + (-0.100 + 0.994i)T^{2} \)
89 \( 1 + (0.928 + 0.850i)T + (0.0875 + 0.996i)T^{2} \)
97 \( 1 + (-1.07 - 0.461i)T + (0.690 + 0.723i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.560492625495211655858925640262, −7.84687592182339315187143227506, −7.40244772228740974837021354265, −6.70860004626346907956628943413, −5.83183692548710058972677561863, −4.84321417293129530672800703622, −3.53460325042992813422826130288, −2.85063022501907553946116931255, −1.93036719131857850545175104719, −1.00398452493991750531598931553, 1.10374475734589034239453554818, 2.27439800071963445146154644090, 3.32175471989567821033753314824, 3.99212014121841335353927630730, 5.23297478161031888762993537966, 5.85866849853650476420747012153, 6.75980129056238970583545234678, 7.53418957694539748343601270617, 8.166332339087536230715661057550, 8.768281906250077524634863068412

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