Properties

Label 2-1656-1.1-c1-0-1
Degree $2$
Conductor $1656$
Sign $1$
Analytic cond. $13.2232$
Root an. cond. $3.63637$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.35·5-s − 2.96·7-s − 1.61·11-s + 2·13-s − 4.96·17-s − 1.35·19-s + 23-s + 6.22·25-s + 7.92·29-s + 5.92·31-s + 9.92·35-s − 2.31·37-s + 1.22·41-s − 4.57·43-s − 1.92·47-s + 1.77·49-s + 4.12·53-s + 5.40·55-s − 2.70·59-s + 14.3·61-s − 6.70·65-s + 4.57·67-s + 16.6·71-s + 2·73-s + 4.77·77-s − 5.03·79-s + 15.0·83-s + ⋯
L(s)  = 1  − 1.49·5-s − 1.11·7-s − 0.486·11-s + 0.554·13-s − 1.20·17-s − 0.309·19-s + 0.208·23-s + 1.24·25-s + 1.47·29-s + 1.06·31-s + 1.67·35-s − 0.380·37-s + 0.191·41-s − 0.697·43-s − 0.280·47-s + 0.253·49-s + 0.566·53-s + 0.728·55-s − 0.351·59-s + 1.83·61-s − 0.831·65-s + 0.558·67-s + 1.97·71-s + 0.234·73-s + 0.544·77-s − 0.566·79-s + 1.64·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1656\)    =    \(2^{3} \cdot 3^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(13.2232\)
Root analytic conductor: \(3.63637\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1656,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7697082689\)
\(L(\frac12)\) \(\approx\) \(0.7697082689\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23 \( 1 - T \)
good5 \( 1 + 3.35T + 5T^{2} \)
7 \( 1 + 2.96T + 7T^{2} \)
11 \( 1 + 1.61T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 4.96T + 17T^{2} \)
19 \( 1 + 1.35T + 19T^{2} \)
29 \( 1 - 7.92T + 29T^{2} \)
31 \( 1 - 5.92T + 31T^{2} \)
37 \( 1 + 2.31T + 37T^{2} \)
41 \( 1 - 1.22T + 41T^{2} \)
43 \( 1 + 4.57T + 43T^{2} \)
47 \( 1 + 1.92T + 47T^{2} \)
53 \( 1 - 4.12T + 53T^{2} \)
59 \( 1 + 2.70T + 59T^{2} \)
61 \( 1 - 14.3T + 61T^{2} \)
67 \( 1 - 4.57T + 67T^{2} \)
71 \( 1 - 16.6T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 5.03T + 79T^{2} \)
83 \( 1 - 15.0T + 83T^{2} \)
89 \( 1 - 5.73T + 89T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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

−9.270875022616642954554970282271, −8.407127600607740410107958171594, −7.973494313291267645424234431169, −6.71302597228446650949557981413, −6.57501418054627492575418375420, −5.11370103577305735513057210660, −4.20830482515485365725303615097, −3.47785657698841285526821800752, −2.55986016065240512488786061072, −0.58374185399782108741194283634, 0.58374185399782108741194283634, 2.55986016065240512488786061072, 3.47785657698841285526821800752, 4.20830482515485365725303615097, 5.11370103577305735513057210660, 6.57501418054627492575418375420, 6.71302597228446650949557981413, 7.973494313291267645424234431169, 8.407127600607740410107958171594, 9.270875022616642954554970282271

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