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Properties

Label 2-6e4-9.5-c2-0-31
Degree $2$
Conductor $1296$
Sign $0.984 - 0.173i$
Analytic cond. $35.3134$
Root an. cond. $5.94251$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.59 + 1.5i)5-s + (2.5 + 4.33i)7-s + (12.9 − 7.5i)11-s + (5 − 8.66i)13-s + 18i·17-s + 16·19-s + (−10.3 − 6i)23-s + (−8 − 13.8i)25-s + (25.9 − 15i)29-s + (−0.5 + 0.866i)31-s + 15.0i·35-s + 20·37-s + (−51.9 − 30i)41-s + (25 + 43.3i)43-s + (5.19 − 3i)47-s + ⋯
L(s)  = 1  + (0.519 + 0.300i)5-s + (0.357 + 0.618i)7-s + (1.18 − 0.681i)11-s + (0.384 − 0.666i)13-s + 1.05i·17-s + 0.842·19-s + (−0.451 − 0.260i)23-s + (−0.320 − 0.554i)25-s + (0.895 − 0.517i)29-s + (−0.0161 + 0.0279i)31-s + 0.428i·35-s + 0.540·37-s + (−1.26 − 0.731i)41-s + (0.581 + 1.00i)43-s + (0.110 − 0.0638i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(35.3134\)
Root analytic conductor: \(5.94251\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.604710625\)
\(L(\frac12)\) \(\approx\) \(2.604710625\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + (-2.59 - 1.5i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-2.5 - 4.33i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-12.9 + 7.5i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-5 + 8.66i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 18iT - 289T^{2} \)
19 \( 1 - 16T + 361T^{2} \)
23 \( 1 + (10.3 + 6i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-25.9 + 15i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 20T + 1.36e3T^{2} \)
41 \( 1 + (51.9 + 30i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-25 - 43.3i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-5.19 + 3i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + 27iT - 2.80e3T^{2} \)
59 \( 1 + (25.9 + 15i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-38 - 65.8i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 90iT - 5.04e3T^{2} \)
73 \( 1 - 65T + 5.32e3T^{2} \)
79 \( 1 + (-7 - 12.1i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (2.59 - 1.5i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 90iT - 7.92e3T^{2} \)
97 \( 1 + (-42.5 - 73.6i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.542769938768664406787177445439, −8.512274438505362387924960664409, −8.174439350565337852717285286215, −6.84492892563719881258293361488, −6.06154646383000165886231531152, −5.55873470715647636152091894370, −4.25613303210661943390122197260, −3.30972193703313582882763878897, −2.18279151986134557095375454037, −1.01246967045697563236866788292, 1.03215548731768872049755561227, 1.89480379202783009763781518664, 3.37700971819589976222083388627, 4.37053068359310132844480949114, 5.10262286958523075505144116094, 6.20889394798299761116221362802, 7.00969105150874114901224553008, 7.68164770478863738366148663561, 8.857305971636273004375812709412, 9.436257284103641561707782261155

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