↑

Properties

Label 2-1344-112.27-c1-0-3
Degree $2$
Conductor $1344$
Sign $-0.745 - 0.666i$
Analytic cond. $10.7318$
Root an. cond. $3.27595$
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  + (0.707 − 0.707i)3-s + (0.308 − 0.308i)5-s + (−1.19 + 2.35i)7-s − 1.00i·9-s + (−2.04 + 2.04i)11-s + (−3.42 − 3.42i)13-s − 0.436i·15-s + 1.82i·17-s + (−3.59 + 3.59i)19-s + (0.821 + 2.51i)21-s − 8.35·23-s + 4.80i·25-s + (−0.707 − 0.707i)27-s + (−0.297 + 0.297i)29-s + 5.96·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.138 − 0.138i)5-s + (−0.452 + 0.891i)7-s − 0.333i·9-s + (−0.617 + 0.617i)11-s + (−0.948 − 0.948i)13-s − 0.112i·15-s + 0.443i·17-s + (−0.823 + 0.823i)19-s + (0.179 + 0.548i)21-s − 1.74·23-s + 0.961i·25-s + (−0.136 − 0.136i)27-s + (−0.0551 + 0.0551i)29-s + 1.07·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.745 - 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.745 - 0.666i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (1231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.745 - 0.666i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4984357311\)
\(L(\frac12)\) \(\approx\) \(0.4984357311\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + (1.19 - 2.35i)T \)
good5 \( 1 + (-0.308 + 0.308i)T - 5iT^{2} \)
11 \( 1 + (2.04 - 2.04i)T - 11iT^{2} \)
13 \( 1 + (3.42 + 3.42i)T + 13iT^{2} \)
17 \( 1 - 1.82iT - 17T^{2} \)
19 \( 1 + (3.59 - 3.59i)T - 19iT^{2} \)
23 \( 1 + 8.35T + 23T^{2} \)
29 \( 1 + (0.297 - 0.297i)T - 29iT^{2} \)
31 \( 1 - 5.96T + 31T^{2} \)
37 \( 1 + (4.17 + 4.17i)T + 37iT^{2} \)
41 \( 1 - 6.03T + 41T^{2} \)
43 \( 1 + (3.98 - 3.98i)T - 43iT^{2} \)
47 \( 1 + 4.18T + 47T^{2} \)
53 \( 1 + (7.54 + 7.54i)T + 53iT^{2} \)
59 \( 1 + (0.385 + 0.385i)T + 59iT^{2} \)
61 \( 1 + (-6.18 - 6.18i)T + 61iT^{2} \)
67 \( 1 + (-2.13 - 2.13i)T + 67iT^{2} \)
71 \( 1 + 6.57T + 71T^{2} \)
73 \( 1 + 3.68T + 73T^{2} \)
79 \( 1 - 6.40iT - 79T^{2} \)
83 \( 1 + (-7.68 + 7.68i)T - 83iT^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 - 15.3iT - 97T^{2} \)
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

−9.959368060158093012833893779402, −9.115334338504692513501597279617, −8.104797165204738015043922505445, −7.78262798118845250066012909527, −6.58617295739427396630055901161, −5.82213064656698410700539575488, −5.00606352924010714101167326890, −3.75072278876554343001418507294, −2.64149812698850934144033530915, −1.88521246396450841844661429847, 0.17406677021574491480802635838, 2.15183461064514493896753656291, 3.07293526070243665445868791844, 4.22121778944368163974157378613, 4.78424682967310302945513995416, 6.13385239127848534936169453439, 6.82896671351842125466640013915, 7.75885773918293341386901019558, 8.476645877811559101846054547804, 9.467251965676156064615155159359

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