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Properties

Label 2-6e4-36.31-c0-0-4
Degree $2$
Conductor $1296$
Sign $0.642 + 0.766i$
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  + (0.866 − 1.5i)5-s + (−0.5 + 0.866i)13-s + 1.73·17-s + (−1 − 1.73i)25-s + (−0.866 − 1.5i)29-s − 37-s + (−0.5 + 0.866i)49-s + (0.5 + 0.866i)61-s + (0.866 + 1.5i)65-s + 73-s + (1.49 − 2.59i)85-s − 1.73·89-s + (1 + 1.73i)97-s + 109-s + (−0.866 + 1.5i)113-s + ⋯
L(s)  = 1  + (0.866 − 1.5i)5-s + (−0.5 + 0.866i)13-s + 1.73·17-s + (−1 − 1.73i)25-s + (−0.866 − 1.5i)29-s − 37-s + (−0.5 + 0.866i)49-s + (0.5 + 0.866i)61-s + (0.866 + 1.5i)65-s + 73-s + (1.49 − 2.59i)85-s − 1.73·89-s + (1 + 1.73i)97-s + 109-s + (−0.866 + 1.5i)113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(0.646788\)
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: \((2,\ 1296,\ (\ :0),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.237819995\)
\(L(\frac12)\) \(\approx\) \(1.237819995\)
\(L(1)\) 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 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - 1.73T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + 1.73T + T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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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

−9.654038949451473267336365359423, −9.079360116296063227949245775843, −8.212118291607557957659815944271, −7.43263097822399704696112342173, −6.20584217763466953136125235390, −5.48292578068894330448260635027, −4.79299073735367316593302660181, −3.79558260551094684447721289707, −2.26016875708946473586186163922, −1.22017105062711633069022253099, 1.76630983033274205211425394897, 2.97941722273514921281687391321, 3.50040371045878046283826747277, 5.24584992237821331603959406813, 5.72485485024873882484510372406, 6.78862306380188653662472231228, 7.34588822667912588295969994849, 8.225001640664484895259892976822, 9.450539852755793507136882068023, 10.06185263315588444666421999513

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