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Properties

Label 2-6e4-9.7-c1-0-20
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

\(L(1)\) \(\approx\) \(1.389247429\)
\(L(\frac12)\) \(\approx\) \(1.389247429\)
\(L(\frac{3}{2})\) 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

−9.349616137934282788151949832386, −8.631773009216457655289104148310, −7.80505830461215580922561570875, −6.90338699782206882538814898869, −6.25634209951146386031091629133, −5.17144354159188181379688716291, −4.20232803554501897870850557385, −3.30706477490523308440198843026, −2.05622063501945365174930143288, −0.55381338827197526572741634589, 1.81070112000778147368059967004, 2.31488220422973967779953729488, 4.13802191171115778905610912905, 4.70485412316276148733355642903, 5.60418252433387890042564675962, 6.77784400855752444409487585816, 7.19065266565793287183054619160, 8.665264483932739708543942344117, 8.947918434215541808881819641105, 9.620884626245311715179437901058

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