↑

Properties

Label 2-1680-105.104-c1-0-24
Degree $2$
Conductor $1680$
Sign $0.999 - 0.0342i$
Analytic cond. $13.4148$
Root an. cond. $3.66263$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.68 − 0.420i)3-s + (−1.08 − 1.95i)5-s + (0.595 − 2.57i)7-s + (2.64 + 1.41i)9-s + 2.82i·11-s − 3.36·13-s + (1 + 3.74i)15-s + 4.75i·17-s + 5.59i·19-s + (−2.08 + 4.08i)21-s + 7.29·23-s + (−2.64 + 4.24i)25-s + (−3.85 − 3.48i)27-s + 0.500i·29-s − 3.06i·31-s + ⋯
L(s)  = 1  + (−0.970 − 0.242i)3-s + (−0.485 − 0.874i)5-s + (0.224 − 0.974i)7-s + (0.881 + 0.471i)9-s + 0.852i·11-s − 0.931·13-s + (0.258 + 0.966i)15-s + 1.15i·17-s + 1.28i·19-s + (−0.454 + 0.890i)21-s + 1.52·23-s + (−0.529 + 0.848i)25-s + (−0.740 − 0.671i)27-s + 0.0930i·29-s − 0.551i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1680\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.999 - 0.0342i$
Analytic conductor: \(13.4148\)
Root analytic conductor: \(3.66263\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1680} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1680,\ (\ :1/2),\ 0.999 - 0.0342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9493148772\)
\(L(\frac12)\) \(\approx\) \(0.9493148772\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.68 + 0.420i)T \)
5 \( 1 + (1.08 + 1.95i)T \)
7 \( 1 + (-0.595 + 2.57i)T \)
good11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 + 3.36T + 13T^{2} \)
17 \( 1 - 4.75iT - 17T^{2} \)
19 \( 1 - 5.59iT - 19T^{2} \)
23 \( 1 - 7.29T + 23T^{2} \)
29 \( 1 - 0.500iT - 29T^{2} \)
31 \( 1 + 3.06iT - 31T^{2} \)
37 \( 1 + 3.32iT - 37T^{2} \)
41 \( 1 - 4.33T + 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + 7.82iT - 47T^{2} \)
53 \( 1 - 8.58T + 53T^{2} \)
59 \( 1 - 2.16T + 59T^{2} \)
61 \( 1 - 2.52iT - 61T^{2} \)
67 \( 1 + 10.3iT - 67T^{2} \)
71 \( 1 - 9.81iT - 71T^{2} \)
73 \( 1 + 5.53T + 73T^{2} \)
79 \( 1 + 3.29T + 79T^{2} \)
83 \( 1 - 6.97iT - 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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

−9.555534003771671281447000807487, −8.386980095667766700595585436614, −7.55537932903088850102055848542, −7.17067649563567283581897971911, −6.07690516503969817485355222859, −5.12971706390745251544541992092, −4.49258465232675933162134089431, −3.78438805713797380521069709629, −1.88245351401774413720950795154, −0.886801822636326176016616652855, 0.57442497553050712370669438346, 2.50967071401429552543348024628, 3.24524557230509253905477240553, 4.64052140509536252130231802104, 5.19531320932414336571346168251, 6.09970675474165049042773857911, 6.99400842299123219517301582825, 7.43229641958710393717519495370, 8.752605624033668810796250457696, 9.315653538629912613289689027954

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