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Properties

Label 2-6e4-9.5-c2-0-33
Degree $2$
Conductor $1296$
Sign $0.996 + 0.0871i$
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  + (1.34 + 0.776i)5-s + (6.19 + 10.7i)7-s + (12.7 − 7.34i)11-s + (5.40 − 9.35i)13-s − 28.9i·17-s − 3.60·19-s + (12.7 + 7.34i)23-s + (−11.2 − 19.5i)25-s + (24.3 − 14.0i)29-s + (4 − 6.92i)31-s + 19.2i·35-s + 22.5·37-s + (−21.7 − 12.5i)41-s + (−26.5 − 46.0i)43-s + (14.6 − 8.48i)47-s + ⋯
L(s)  = 1  + (0.268 + 0.155i)5-s + (0.885 + 1.53i)7-s + (1.15 − 0.668i)11-s + (0.415 − 0.719i)13-s − 1.70i·17-s − 0.189·19-s + (0.553 + 0.319i)23-s + (−0.451 − 0.782i)25-s + (0.840 − 0.485i)29-s + (0.129 − 0.223i)31-s + 0.549i·35-s + 0.609·37-s + (−0.531 − 0.306i)41-s + (−0.618 − 1.07i)43-s + (0.312 − 0.180i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0871i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 + 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $0.996 + 0.0871i$
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: \((2,\ 1296,\ (\ :1),\ 0.996 + 0.0871i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.605421898\)
\(L(\frac12)\) \(\approx\) \(2.605421898\)
\(L(2)\) 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 \( 1 + (-1.34 - 0.776i)T + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-6.19 - 10.7i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (-12.7 + 7.34i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-5.40 + 9.35i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + 28.9iT - 289T^{2} \)
19 \( 1 + 3.60T + 361T^{2} \)
23 \( 1 + (-12.7 - 7.34i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (-24.3 + 14.0i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 - 22.5T + 1.36e3T^{2} \)
41 \( 1 + (21.7 + 12.5i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (26.5 + 46.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-14.6 + 8.48i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 84.5iT - 2.80e3T^{2} \)
59 \( 1 + (-78.8 - 45.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (20.5 - 35.6i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 16.3iT - 5.04e3T^{2} \)
73 \( 1 - 71.5T + 5.32e3T^{2} \)
79 \( 1 + (23.3 + 40.4i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (13.2 - 7.65i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 78.9iT - 7.92e3T^{2} \)
97 \( 1 + (45.5 + 78.9i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.229320555740110934788981404290, −8.743355931276315921932674098287, −8.026379298235067705394138756658, −6.91792708910938120540404277740, −5.95363278473194566220542216172, −5.41588220410948527186987258231, −4.41227254141978231466633873672, −3.07354869556069022659905132477, −2.25669068901246197947585549699, −0.911952933556152353326940429159, 1.19547331783757989666477646572, 1.76440378832706522845086197840, 3.68161853251145563280570648252, 4.22257797608243612383203948187, 5.06052531519791338303499925765, 6.48760550412640781903401489203, 6.82750224425327957605852742430, 7.937529695198057740366208304529, 8.562347050098573066730960310978, 9.569888983314554766248482933082

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