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Properties

Label 2-6e4-9.7-c1-0-15
Degree $2$
Conductor $1296$
Sign $-0.173 + 0.984i$
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  + (−0.5 − 0.866i)5-s + (1.5 − 2.59i)7-s + (−2.5 + 4.33i)11-s + (−2 − 3.46i)13-s + 8·17-s − 2·19-s + (−1 − 1.73i)23-s + (2 − 3.46i)25-s + (3 − 5.19i)29-s + (−3.5 − 6.06i)31-s − 3·35-s − 6·37-s + (−3 − 5.19i)41-s + (−1 + 1.73i)43-s + (−3 + 5.19i)47-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.566 − 0.981i)7-s + (−0.753 + 1.30i)11-s + (−0.554 − 0.960i)13-s + 1.94·17-s − 0.458·19-s + (−0.208 − 0.361i)23-s + (0.400 − 0.692i)25-s + (0.557 − 0.964i)29-s + (−0.628 − 1.08i)31-s − 0.507·35-s − 0.986·37-s + (−0.468 − 0.811i)41-s + (−0.152 + 0.264i)43-s + (−0.437 + 0.757i)47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(10.3486\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ -0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.310097070\)
\(L(\frac12)\) \(\approx\) \(1.310097070\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 8T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 6T + 37T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5T + 53T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (-8 + 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.5 + 9.52i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.708237666229638513035335658785, −8.358138279158173895697688255400, −7.68461560325797241443749600563, −7.37229100983517504683484701326, −6.01913271044382748649558146395, −4.97871290712135724249298534418, −4.46664127299877456505881493545, −3.28774685604053278403711470721, −1.98682673528538628552122448208, −0.55650432350920045165965023985, 1.53298576298841496429823686559, 2.85688251941029706379595922295, 3.57349574528305212738691426525, 5.16702226778786525436816693439, 5.43479583850992930772514428893, 6.61978818782588721857087892276, 7.49614286303348320202468048615, 8.381845887463191624501768673575, 8.857507926843397652267215255566, 9.944024113216631858437075179362

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