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Properties

Label 2-6e4-9.5-c2-0-12
Degree $2$
Conductor $1296$
Sign $-0.642 - 0.766i$
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  + (5.5 + 9.52i)7-s + (−11.5 + 19.9i)13-s + 37·19-s + (−12.5 − 21.6i)25-s + (−23 + 39.8i)31-s − 73·37-s + (−11 − 19.0i)43-s + (−36 + 62.3i)49-s + (−23.5 − 40.7i)61-s + (−6.5 + 11.2i)67-s + 143·73-s + (5.5 + 9.52i)79-s − 253·91-s + (84.5 + 146. i)97-s + (−78.5 + 135. i)103-s + ⋯
L(s)  = 1  + (0.785 + 1.36i)7-s + (−0.884 + 1.53i)13-s + 1.94·19-s + (−0.5 − 0.866i)25-s + (−0.741 + 1.28i)31-s − 1.97·37-s + (−0.255 − 0.443i)43-s + (−0.734 + 1.27i)49-s + (−0.385 − 0.667i)61-s + (−0.0970 + 0.168i)67-s + 1.95·73-s + (0.0696 + 0.120i)79-s − 2.78·91-s + (0.871 + 1.50i)97-s + (−0.762 + 1.32i)103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $-0.642 - 0.766i$
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: \((2,\ 1296,\ (\ :1),\ -0.642 - 0.766i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.516859289\)
\(L(\frac12)\) \(\approx\) \(1.516859289\)
\(L(2)\) 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 \( 1 + (12.5 + 21.6i)T^{2} \)
7 \( 1 + (-5.5 - 9.52i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (60.5 - 104. i)T^{2} \)
13 \( 1 + (11.5 - 19.9i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 - 37T + 361T^{2} \)
23 \( 1 + (264.5 + 458. i)T^{2} \)
29 \( 1 + (420.5 - 728. i)T^{2} \)
31 \( 1 + (23 - 39.8i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 73T + 1.36e3T^{2} \)
41 \( 1 + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (11 + 19.0i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (23.5 + 40.7i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (6.5 - 11.2i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 143T + 5.32e3T^{2} \)
79 \( 1 + (-5.5 - 9.52i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + (-84.5 - 146. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.491916162995338632485745474389, −9.068009341495681843581214566932, −8.215366034070726822555281767790, −7.31323916647783776161616920568, −6.51286214801338388695992446768, −5.26132808146977073078331134908, −5.01609126649749524861272647829, −3.64272914534569088635174482186, −2.42164441385462578075741634396, −1.59850011903782313259005445872, 0.43424280624175943825081437232, 1.54521801179125261400485037664, 3.05806191203817950983263147054, 3.89960281041964492089146350979, 5.07720104268189621158030876901, 5.52882351825582088230558306108, 7.02542777526991203048361344137, 7.60258845307742530676905928177, 8.025182240327055554162865718635, 9.348533940456258489932083887323

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