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Properties

Label 2-1344-16.13-c1-0-9
Degree $2$
Conductor $1344$
Sign $-0.312 - 0.950i$
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  + (0.707 + 0.707i)3-s + (−1.17 + 1.17i)5-s + i·7-s + 1.00i·9-s + (0.961 − 0.961i)11-s + (2.26 + 2.26i)13-s − 1.65·15-s + 3.43·17-s + (−0.568 − 0.568i)19-s + (−0.707 + 0.707i)21-s − 2.04i·23-s + 2.24i·25-s + (−0.707 + 0.707i)27-s + (6.38 + 6.38i)29-s − 9.65·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−0.524 + 0.524i)5-s + 0.377i·7-s + 0.333i·9-s + (0.289 − 0.289i)11-s + (0.626 + 0.626i)13-s − 0.428·15-s + 0.832·17-s + (−0.130 − 0.130i)19-s + (−0.154 + 0.154i)21-s − 0.426i·23-s + 0.449i·25-s + (−0.136 + 0.136i)27-s + (1.18 + 1.18i)29-s − 1.73·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.312 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.312 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.312 - 0.950i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.312 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.583094123\)
\(L(\frac12)\) \(\approx\) \(1.583094123\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good5 \( 1 + (1.17 - 1.17i)T - 5iT^{2} \)
11 \( 1 + (-0.961 + 0.961i)T - 11iT^{2} \)
13 \( 1 + (-2.26 - 2.26i)T + 13iT^{2} \)
17 \( 1 - 3.43T + 17T^{2} \)
19 \( 1 + (0.568 + 0.568i)T + 19iT^{2} \)
23 \( 1 + 2.04iT - 23T^{2} \)
29 \( 1 + (-6.38 - 6.38i)T + 29iT^{2} \)
31 \( 1 + 9.65T + 31T^{2} \)
37 \( 1 + (8.01 - 8.01i)T - 37iT^{2} \)
41 \( 1 + 0.877iT - 41T^{2} \)
43 \( 1 + (1.46 - 1.46i)T - 43iT^{2} \)
47 \( 1 - 0.509T + 47T^{2} \)
53 \( 1 + (4.42 - 4.42i)T - 53iT^{2} \)
59 \( 1 + (1.96 - 1.96i)T - 59iT^{2} \)
61 \( 1 + (-4.61 - 4.61i)T + 61iT^{2} \)
67 \( 1 + (0.0452 + 0.0452i)T + 67iT^{2} \)
71 \( 1 - 9.10iT - 71T^{2} \)
73 \( 1 + 9.52iT - 73T^{2} \)
79 \( 1 + 4.83T + 79T^{2} \)
83 \( 1 + (-9.73 - 9.73i)T + 83iT^{2} \)
89 \( 1 + 12.5iT - 89T^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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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

−9.837810498093899529569249029697, −8.869780044813427162299378224345, −8.482874933124482362025859252357, −7.39463238579422522627978013361, −6.69960236649623553371977634874, −5.66949866023431208991685309206, −4.68586231756219301677030805186, −3.60135036570776783516835284370, −3.05305698074603189370463430650, −1.58110886857661959730285902551, 0.64661954996316349871135275246, 1.90070617990074858351575155172, 3.35490114631901332469786887715, 4.01563120501828017752783308059, 5.14775150515722183381364961221, 6.10047109719926333615613407597, 7.09619909966948622937565248082, 7.85297488182000572428446496032, 8.408315492094200895782571019327, 9.259231886552577483602153934862

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