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Properties

Label 2-6e4-9.4-c1-0-11
Degree $2$
Conductor $1296$
Sign $0.939 + 0.342i$
Analytic cond. $10.3486$
Root an. cond. $3.21692$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 − 1.73i)5-s + (2 + 3.46i)11-s + (1 − 1.73i)13-s + 2·17-s + 4·19-s + (−4 + 6.92i)23-s + (0.500 + 0.866i)25-s + (−3 − 5.19i)29-s + (4 − 6.92i)31-s + 6·37-s + (3 − 5.19i)41-s + (2 + 3.46i)43-s + (3.5 − 6.06i)49-s − 2·53-s + 7.99·55-s + ⋯
L(s)  = 1  + (0.447 − 0.774i)5-s + (0.603 + 1.04i)11-s + (0.277 − 0.480i)13-s + 0.485·17-s + 0.917·19-s + (−0.834 + 1.44i)23-s + (0.100 + 0.173i)25-s + (−0.557 − 0.964i)29-s + (0.718 − 1.24i)31-s + 0.986·37-s + (0.468 − 0.811i)41-s + (0.304 + 0.528i)43-s + (0.5 − 0.866i)49-s − 0.274·53-s + 1.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(10.3486\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.946536842\)
\(L(\frac12)\) \(\approx\) \(1.946536842\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (4 - 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2 + 3.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.693080612129533857606690196559, −8.983782134721568185013332611342, −7.85386502371059481741805053357, −7.37971193477624688681199347483, −6.06476310441097639881411943699, −5.50723316545421681844032654982, −4.49322110001952021370024627619, −3.59357217927929828770270182953, −2.15828077194064114769502294055, −1.06535465090911142508717297971, 1.15379523108787770338247329142, 2.61186409721887211558019447469, 3.43926254724106768493647356392, 4.52768910472864278499460128736, 5.75833180442213004100776537015, 6.35430308442182449281840083855, 7.09746766904641840388238040625, 8.162894979831527448249214535997, 8.897857964504639591692999712761, 9.739436916257339705893562759834

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