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Properties

Label 2-1344-7.2-c1-0-22
Degree $2$
Conductor $1344$
Sign $0.701 + 0.712i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (−1 + 1.73i)5-s + (−0.5 − 2.59i)7-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)11-s + 3·13-s − 1.99·15-s + (−4 − 6.92i)17-s + (0.5 − 0.866i)19-s + (2 − 1.73i)21-s + (4 − 6.92i)23-s + (0.500 + 0.866i)25-s − 0.999·27-s − 4·29-s + (1.5 + 2.59i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.447 + 0.774i)5-s + (−0.188 − 0.981i)7-s + (−0.166 + 0.288i)9-s + (−0.301 − 0.522i)11-s + 0.832·13-s − 0.516·15-s + (−0.970 − 1.68i)17-s + (0.114 − 0.198i)19-s + (0.436 − 0.377i)21-s + (0.834 − 1.44i)23-s + (0.100 + 0.173i)25-s − 0.192·27-s − 0.742·29-s + (0.269 + 0.466i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.376491942\)
\(L(\frac12)\) \(\approx\) \(1.376491942\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 2.59i)T \)
good5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (4 + 6.92i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 11T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.5 - 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.442762958832870556906392577182, −8.819841936814681687881718319451, −7.81989764136959506018270215293, −7.06132097135138940483671838861, −6.44930162662236058896372068894, −5.12158109187847845390881937307, −4.25352605856111160578388606397, −3.35917637416682304591204486230, −2.61015082609295694800407186166, −0.58723040207246111268741043282, 1.33640549561739364456633046631, 2.40446399393682238116595739071, 3.65964175278412240896256917411, 4.55497643096616142477776282294, 5.73913473769335494991801826999, 6.25450514381524419861371149406, 7.52529203124081862318527096112, 8.085561545890165162483139515352, 9.074055141679470985004643462041, 9.180764983506823298087412908742

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