Properties

Label 2-1344-112.27-c1-0-12
Degree $2$
Conductor $1344$
Sign $0.706 - 0.707i$
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 + (1.02 − 1.02i)5-s + (−2.25 + 1.37i)7-s − 1.00i·9-s + (0.872 − 0.872i)11-s + (2.13 + 2.13i)13-s + 1.44i·15-s − 2.47i·17-s + (0.479 − 0.479i)19-s + (0.622 − 2.57i)21-s + 0.386·23-s + 2.89i·25-s + (0.707 + 0.707i)27-s + (−2.27 + 2.27i)29-s + 10.5·31-s + ⋯
L(s)  = 1  + (−0.408 + 0.408i)3-s + (0.458 − 0.458i)5-s + (−0.853 + 0.520i)7-s − 0.333i·9-s + (0.263 − 0.263i)11-s + (0.591 + 0.591i)13-s + 0.374i·15-s − 0.600i·17-s + (0.110 − 0.110i)19-s + (0.135 − 0.561i)21-s + 0.0805·23-s + 0.579i·25-s + (0.136 + 0.136i)27-s + (−0.422 + 0.422i)29-s + 1.89·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.706 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.706 - 0.707i$
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: \((2,\ 1344,\ (\ :1/2),\ 0.706 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.409976756\)
\(L(\frac12)\) \(\approx\) \(1.409976756\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.25 - 1.37i)T \)
good5 \( 1 + (-1.02 + 1.02i)T - 5iT^{2} \)
11 \( 1 + (-0.872 + 0.872i)T - 11iT^{2} \)
13 \( 1 + (-2.13 - 2.13i)T + 13iT^{2} \)
17 \( 1 + 2.47iT - 17T^{2} \)
19 \( 1 + (-0.479 + 0.479i)T - 19iT^{2} \)
23 \( 1 - 0.386T + 23T^{2} \)
29 \( 1 + (2.27 - 2.27i)T - 29iT^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 + (-2.46 - 2.46i)T + 37iT^{2} \)
41 \( 1 - 0.121T + 41T^{2} \)
43 \( 1 + (6.98 - 6.98i)T - 43iT^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 + (3.51 + 3.51i)T + 53iT^{2} \)
59 \( 1 + (-9.73 - 9.73i)T + 59iT^{2} \)
61 \( 1 + (-8.19 - 8.19i)T + 61iT^{2} \)
67 \( 1 + (7.20 + 7.20i)T + 67iT^{2} \)
71 \( 1 - 11.4T + 71T^{2} \)
73 \( 1 + 3.95T + 73T^{2} \)
79 \( 1 - 8.75iT - 79T^{2} \)
83 \( 1 + (3.89 - 3.89i)T - 83iT^{2} \)
89 \( 1 - 8.14T + 89T^{2} \)
97 \( 1 + 0.654iT - 97T^{2} \)
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

−9.631462246748802325478618622487, −9.077361441565292171957335607854, −8.371333000097662254234224659480, −7.02341113449625795235170053562, −6.30169278338100986988274905785, −5.59449107686603236327935425024, −4.72482652438257796128047665028, −3.67201961146173798453876445917, −2.62858718747621791881659946787, −1.08442264298433014709429442035, 0.75983135327990431832120610201, 2.21264839892868351589253229276, 3.34222401396003606757316903905, 4.31467125211502502436054803044, 5.60458834419065069669870998052, 6.31775293366079486301459261749, 6.83526881007734873021026614456, 7.80417885563434271877066509043, 8.643667966592299804912651911353, 9.796204623440671835436525318628

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