↑

Properties

Label 2-1344-112.27-c1-0-27
Degree $2$
Conductor $1344$
Sign $0.249 + 0.968i$
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 + (2.46 − 2.46i)5-s + (2.50 + 0.865i)7-s − 1.00i·9-s + (−0.244 + 0.244i)11-s + (−2.35 − 2.35i)13-s − 3.48i·15-s − 5.19i·17-s + (−1.43 + 1.43i)19-s + (2.37 − 1.15i)21-s + 6.61·23-s − 7.16i·25-s + (−0.707 − 0.707i)27-s + (−5.09 + 5.09i)29-s + 1.53·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (1.10 − 1.10i)5-s + (0.945 + 0.326i)7-s − 0.333i·9-s + (−0.0737 + 0.0737i)11-s + (−0.652 − 0.652i)13-s − 0.900i·15-s − 1.25i·17-s + (−0.328 + 0.328i)19-s + (0.519 − 0.252i)21-s + 1.37·23-s − 1.43i·25-s + (−0.136 − 0.136i)27-s + (−0.945 + 0.945i)29-s + 0.275·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.249 + 0.968i$
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: \((2,\ 1344,\ (\ :1/2),\ 0.249 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.504066861\)
\(L(\frac12)\) \(\approx\) \(2.504066861\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.50 - 0.865i)T \)
good5 \( 1 + (-2.46 + 2.46i)T - 5iT^{2} \)
11 \( 1 + (0.244 - 0.244i)T - 11iT^{2} \)
13 \( 1 + (2.35 + 2.35i)T + 13iT^{2} \)
17 \( 1 + 5.19iT - 17T^{2} \)
19 \( 1 + (1.43 - 1.43i)T - 19iT^{2} \)
23 \( 1 - 6.61T + 23T^{2} \)
29 \( 1 + (5.09 - 5.09i)T - 29iT^{2} \)
31 \( 1 - 1.53T + 31T^{2} \)
37 \( 1 + (2.40 + 2.40i)T + 37iT^{2} \)
41 \( 1 - 9.55T + 41T^{2} \)
43 \( 1 + (6.80 - 6.80i)T - 43iT^{2} \)
47 \( 1 + 7.20T + 47T^{2} \)
53 \( 1 + (-5.49 - 5.49i)T + 53iT^{2} \)
59 \( 1 + (-3.45 - 3.45i)T + 59iT^{2} \)
61 \( 1 + (1.27 + 1.27i)T + 61iT^{2} \)
67 \( 1 + (3.83 + 3.83i)T + 67iT^{2} \)
71 \( 1 + 16.0T + 71T^{2} \)
73 \( 1 - 6.68T + 73T^{2} \)
79 \( 1 - 4.09iT - 79T^{2} \)
83 \( 1 + (-9.70 + 9.70i)T - 83iT^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + 16.7iT - 97T^{2} \)
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

−9.170842382012426965003093978809, −8.857276140324522518575134595915, −7.85911964768562893283499693397, −7.17782796805764784639491306568, −5.94141356474091545946895716825, −5.15437376836327664992905471929, −4.68677702286153577913790626149, −2.98210143965658111820799047285, −2.01260376639583284939947749616, −1.04163752563970340371817526458, 1.76524254419176231764314511184, 2.49917822290034086869805506786, 3.67422628908864519459315426188, 4.69354659501686712184995717694, 5.60035692175617831333146464878, 6.57760104847700912867630546231, 7.27161872878790493507298212335, 8.217456217747977018725615107407, 9.086922012783073559899329771867, 9.853763675549078594161262406466

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