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Properties

Label 2-6e4-9.5-c2-0-0
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  + (−6.5 − 11.2i)7-s + (0.5 − 0.866i)13-s − 11·19-s + (−12.5 − 21.6i)25-s + (−23 + 39.8i)31-s + 47·37-s + (−11 − 19.0i)43-s + (−59.9 + 103. i)49-s + (60.5 + 104. i)61-s + (−54.5 + 94.3i)67-s − 97·73-s + (65.5 + 113. i)79-s − 12.9·91-s + (−83.5 − 144. i)97-s + (−18.5 + 32.0i)103-s + ⋯
L(s)  = 1  + (−0.928 − 1.60i)7-s + (0.0384 − 0.0666i)13-s − 0.578·19-s + (−0.5 − 0.866i)25-s + (−0.741 + 1.28i)31-s + 1.27·37-s + (−0.255 − 0.443i)43-s + (−1.22 + 2.12i)49-s + (0.991 + 1.71i)61-s + (−0.813 + 1.40i)67-s − 1.32·73-s + (0.829 + 1.43i)79-s − 0.142·91-s + (−0.860 − 1.49i)97-s + (−0.179 + 0.311i)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

\(L(\frac{3}{2})\) \(\approx\) \(0.1012881697\)
\(L(\frac12)\) \(\approx\) \(0.1012881697\)
\(L(2)\) 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 \( 1 + (6.5 + 11.2i)T + (-24.5 + 42.4i)T^{2} \)
11 \( 1 + (60.5 - 104. i)T^{2} \)
13 \( 1 + (-0.5 + 0.866i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 - 289T^{2} \)
19 \( 1 + 11T + 361T^{2} \)
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 \( 1 - 47T + 1.36e3T^{2} \)
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 \( 1 + (-60.5 - 104. i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (54.5 - 94.3i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 + 97T + 5.32e3T^{2} \)
79 \( 1 + (-65.5 - 113. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + (83.5 + 144. i)T + (-4.70e3 + 8.14e3i)T^{2} \)
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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

−9.973708604743540965086335375492, −9.004176202495395048045421298372, −8.061029544396139141582433814426, −7.17308505336157637722294690235, −6.64503213971802430032652054959, −5.68781367111092981616130953656, −4.38776303088010611944344675765, −3.81323554035523544175143810644, −2.72677701092446994315697369618, −1.16296145795035519375209442411, 0.03071236791959787649490781463, 1.94080993767876118580718709252, 2.83465129213179932479027971862, 3.83285804115538318324925771124, 5.08733167273317602577875824718, 5.94348905687114331346657746190, 6.44515477515060847351979149442, 7.62044227715444970228970023877, 8.460275830729901603170063166151, 9.405728234474972368846950593356

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