↑

Properties

Label 2-3736-3736.155-c0-0-0
Degree $2$
Conductor $3736$
Sign $-0.372 + 0.928i$
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  + (0.114 − 0.993i)2-s + (0.0940 + 0.349i)3-s + (−0.973 − 0.227i)4-s + (0.358 − 0.0534i)6-s + (−0.337 + 0.941i)8-s + (0.751 − 0.436i)9-s + (0.717 − 1.47i)11-s + (−0.0122 − 0.361i)12-s + (0.896 + 0.442i)16-s + (−0.709 − 1.51i)17-s + (−0.347 − 0.796i)18-s + (−0.680 + 1.75i)19-s + (−1.38 − 0.881i)22-s + (−0.360 − 0.0292i)24-s + (0.999 − 0.0269i)25-s + ⋯
L(s)  = 1  + (0.114 − 0.993i)2-s + (0.0940 + 0.349i)3-s + (−0.973 − 0.227i)4-s + (0.358 − 0.0534i)6-s + (−0.337 + 0.941i)8-s + (0.751 − 0.436i)9-s + (0.717 − 1.47i)11-s + (−0.0122 − 0.361i)12-s + (0.896 + 0.442i)16-s + (−0.709 − 1.51i)17-s + (−0.347 − 0.796i)18-s + (−0.680 + 1.75i)19-s + (−1.38 − 0.881i)22-s + (−0.360 − 0.0292i)24-s + (0.999 − 0.0269i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3736\)    =    \(2^{3} \cdot 467\)
Sign: $-0.372 + 0.928i$
Analytic conductor: \(1.86450\)
Root analytic conductor: \(1.36546\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3736} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3736,\ (\ :0),\ -0.372 + 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.294544537\)
\(L(\frac12)\) \(\approx\) \(1.294544537\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.114 + 0.993i)T \)
467 \( 1 + (-0.948 + 0.317i)T \)
good3 \( 1 + (-0.0940 - 0.349i)T + (-0.864 + 0.501i)T^{2} \)
5 \( 1 + (-0.999 + 0.0269i)T^{2} \)
7 \( 1 + (0.805 - 0.592i)T^{2} \)
11 \( 1 + (-0.717 + 1.47i)T + (-0.618 - 0.785i)T^{2} \)
13 \( 1 + (-0.829 - 0.559i)T^{2} \)
17 \( 1 + (0.709 + 1.51i)T + (-0.639 + 0.768i)T^{2} \)
19 \( 1 + (0.680 - 1.75i)T + (-0.737 - 0.675i)T^{2} \)
23 \( 1 + (0.484 - 0.874i)T^{2} \)
29 \( 1 + (0.127 - 0.991i)T^{2} \)
31 \( 1 + (-0.709 - 0.704i)T^{2} \)
37 \( 1 + (-0.542 + 0.840i)T^{2} \)
41 \( 1 + (-0.00820 + 1.21i)T + (-0.999 - 0.0134i)T^{2} \)
43 \( 1 + (0.114 + 0.166i)T + (-0.362 + 0.932i)T^{2} \)
47 \( 1 + (-0.746 + 0.665i)T^{2} \)
53 \( 1 + (0.311 + 0.950i)T^{2} \)
59 \( 1 + (0.696 + 1.86i)T + (-0.755 + 0.655i)T^{2} \)
61 \( 1 + (-0.272 - 0.962i)T^{2} \)
67 \( 1 + (-1.78 - 0.821i)T + (0.650 + 0.759i)T^{2} \)
71 \( 1 + (0.952 + 0.305i)T^{2} \)
73 \( 1 + (0.671 + 1.04i)T + (-0.412 + 0.911i)T^{2} \)
79 \( 1 + (0.943 - 0.330i)T^{2} \)
83 \( 1 + (-0.930 - 0.556i)T + (0.472 + 0.881i)T^{2} \)
89 \( 1 + (-1.66 - 1.09i)T + (0.399 + 0.916i)T^{2} \)
97 \( 1 + (0.712 + 1.69i)T + (-0.699 + 0.714i)T^{2} \)
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

−8.704074023489138259935633117784, −8.048205341620423434016621183309, −6.87335104198917555844193496796, −6.14980451474033095122307517515, −5.24756005866424090027821972669, −4.42305267836991496915986616870, −3.67373708798700415651902764124, −3.11850471416513347302648104534, −1.90593955802836357972018934901, −0.793417640492699279452120977923, 1.39338772795403323001995851856, 2.45964655452168289806908479976, 3.88034923656636067959555300635, 4.57612656921874799803985213430, 4.96824234157571041598556253219, 6.43763615996667420725013900218, 6.60853572254547366772698595768, 7.31685269683442360413304634578, 8.041682791393583229490477413787, 8.820024655403151234473720567250

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