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Properties

Label 2-6e4-36.31-c0-0-1
Degree $2$
Conductor $1296$
Sign $-0.173 - 0.984i$
Analytic cond. $0.646788$
Root an. cond. $0.804231$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)7-s + (−0.5 + 0.866i)13-s + 1.73i·19-s + (0.5 + 0.866i)25-s − 37-s + (1 − 1.73i)49-s + (0.5 + 0.866i)61-s + (−1.5 − 0.866i)67-s + 73-s + (1.5 − 0.866i)79-s − 1.73i·91-s + (−0.5 − 0.866i)97-s + (1.5 + 0.866i)103-s − 2·109-s + ⋯
L(s)  = 1  + (−1.5 + 0.866i)7-s + (−0.5 + 0.866i)13-s + 1.73i·19-s + (0.5 + 0.866i)25-s − 37-s + (1 − 1.73i)49-s + (0.5 + 0.866i)61-s + (−1.5 − 0.866i)67-s + 73-s + (1.5 − 0.866i)79-s − 1.73i·91-s + (−0.5 − 0.866i)97-s + (1.5 + 0.866i)103-s − 2·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(0.646788\)
Root analytic conductor: \(0.804231\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :0),\ -0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6997494220\)
\(L(\frac12)\) \(\approx\) \(0.6997494220\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - 1.73iT - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.919836076926915501974594132398, −9.311455163150950780164805280022, −8.669060831731531756725945666178, −7.56187268558252173153700921643, −6.67904247366919260201388449221, −6.01671241690362663200240029680, −5.16697932780935952388625824056, −3.86408416361333946066487271286, −3.06804887972638879037669248962, −1.89411824156457209905851467830, 0.57489603736880578017237761974, 2.61472737352510754164452097648, 3.37506936342690733407412296345, 4.44429522553574789214401890130, 5.42565465965336258914538845624, 6.60041019891404903192229002763, 6.95689306953257375087321554908, 7.930610108934299194122012390095, 8.977847681960974324485170616804, 9.674812035895073043306014984345

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