Properties

Label 2-1344-7.4-c1-0-17
Degree $2$
Conductor $1344$
Sign $0.968 + 0.250i$
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 + (0.5 + 0.866i)5-s + (2.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s − 0.999·15-s + (4 − 6.92i)17-s + (−2 − 3.46i)19-s + (−0.500 + 2.59i)21-s + (−2 − 3.46i)23-s + (2 − 3.46i)25-s + 0.999·27-s + 5·29-s + (−3.5 + 6.06i)31-s + (0.499 + 0.866i)33-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.223 + 0.387i)5-s + (0.944 − 0.327i)7-s + (−0.166 − 0.288i)9-s + (0.150 − 0.261i)11-s − 0.258·15-s + (0.970 − 1.68i)17-s + (−0.458 − 0.794i)19-s + (−0.109 + 0.566i)21-s + (−0.417 − 0.722i)23-s + (0.400 − 0.692i)25-s + 0.192·27-s + 0.928·29-s + (−0.628 + 1.08i)31-s + (0.0870 + 0.150i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.731740930\)
\(L(\frac12)\) \(\approx\) \(1.731740930\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 0.866i)T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-4 + 6.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.5 + 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.5 - 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−9.771575565859570022491580385872, −8.742246299390820731005239613668, −8.064522442701276376092597899293, −7.00829318203567106109216455654, −6.36985627647395589409330629028, −5.05290141194100084625959097186, −4.77944480205365351660651605344, −3.46389469796025996116386638006, −2.45067608633037747285680715120, −0.846239260396293073982028686291, 1.32647461576426298406969447795, 2.05058978641858857343839272505, 3.64120534326883049856286020436, 4.63989452383242343099588936017, 5.68887996800110533036229011674, 6.06543630531402533596344589126, 7.38342539216342759655630631574, 8.022795475321647779525766668366, 8.648337938869299425675133949218, 9.659518403218010228369853799927

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