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Properties

Label 2-6e4-4.3-c2-0-38
Degree $2$
Conductor $1296$
Sign $-0.5 + 0.866i$
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  − 0.710·5-s − 3.12i·7-s − 16.6i·11-s + 18.3·13-s − 9.69·17-s + 8.20i·19-s + 2.24i·23-s − 24.4·25-s + 41.6·29-s − 24.9i·31-s + 2.21i·35-s − 40.3·37-s − 51.3·41-s + 65.4i·43-s − 33.8i·47-s + ⋯
L(s)  = 1  − 0.142·5-s − 0.446i·7-s − 1.50i·11-s + 1.41·13-s − 0.570·17-s + 0.431i·19-s + 0.0978i·23-s − 0.979·25-s + 1.43·29-s − 0.805i·31-s + 0.0634i·35-s − 1.09·37-s − 1.25·41-s + 1.52i·43-s − 0.719i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $-0.5 + 0.866i$
Analytic conductor: \(35.3134\)
Root analytic conductor: \(5.94251\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (1135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1),\ -0.5 + 0.866i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.321932974\)
\(L(\frac12)\) \(\approx\) \(1.321932974\)
\(L(2)\) 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 \( 1 + 0.710T + 25T^{2} \)
7 \( 1 + 3.12iT - 49T^{2} \)
11 \( 1 + 16.6iT - 121T^{2} \)
13 \( 1 - 18.3T + 169T^{2} \)
17 \( 1 + 9.69T + 289T^{2} \)
19 \( 1 - 8.20iT - 361T^{2} \)
23 \( 1 - 2.24iT - 529T^{2} \)
29 \( 1 - 41.6T + 841T^{2} \)
31 \( 1 + 24.9iT - 961T^{2} \)
37 \( 1 + 40.3T + 1.36e3T^{2} \)
41 \( 1 + 51.3T + 1.68e3T^{2} \)
43 \( 1 - 65.4iT - 1.84e3T^{2} \)
47 \( 1 + 33.8iT - 2.20e3T^{2} \)
53 \( 1 + 90.6T + 2.80e3T^{2} \)
59 \( 1 + 76.4iT - 3.48e3T^{2} \)
61 \( 1 + 2.71T + 3.72e3T^{2} \)
67 \( 1 + 39.8iT - 4.48e3T^{2} \)
71 \( 1 + 102. iT - 5.04e3T^{2} \)
73 \( 1 - 38.1T + 5.32e3T^{2} \)
79 \( 1 + 109. iT - 6.24e3T^{2} \)
83 \( 1 + 131. iT - 6.88e3T^{2} \)
89 \( 1 - 38.0T + 7.92e3T^{2} \)
97 \( 1 - 24.3T + 9.40e3T^{2} \)
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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

−9.038036691258433205961836208305, −8.358055054379678347907488805227, −7.78451853983982323108417724528, −6.42844935537117542682176474480, −6.12481534909018431693801406738, −4.92342780443338262631012885457, −3.79725233585335470691488924544, −3.20294987935038748360324272147, −1.61370390568652682574062852438, −0.39118830365156032436369365905, 1.41935558666011904672384563856, 2.48597443686113361082687470850, 3.73063751623694121958553366348, 4.60517582007441845411796632916, 5.51751610596810108048938028090, 6.58151370650630936307763018233, 7.13265451951738732269619463051, 8.316855041156827387780583529110, 8.794059312091995201216937349054, 9.757028595756946350282937473959

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