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Properties

Label 2-6e4-12.11-c1-0-11
Degree $2$
Conductor $1296$
Sign $1$
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  + 0.896i·5-s + 3.19·13-s − 8.24i·17-s + 4.19·25-s + 6.45i·29-s + 9.39·37-s + 12.7i·41-s + 7·49-s − 12.7i·53-s + 15.3·61-s + 2.86i·65-s − 2.80·73-s + 7.39·85-s − 13.8i·89-s − 8·97-s + ⋯
L(s)  = 1  + 0.400i·5-s + 0.886·13-s − 1.99i·17-s + 0.839·25-s + 1.19i·29-s + 1.54·37-s + 1.98i·41-s + 49-s − 1.74i·53-s + 1.97·61-s + 0.355i·65-s − 0.328·73-s + 0.801·85-s − 1.46i·89-s − 0.812·97-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $1$
Analytic conductor: \(10.3486\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (1295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.735691755\)
\(L(\frac12)\) \(\approx\) \(1.735691755\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.896iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 3.19T + 13T^{2} \)
17 \( 1 + 8.24iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 6.45iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 9.39T + 37T^{2} \)
41 \( 1 - 12.7iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 12.7iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 15.3T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2.80T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 13.8iT - 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−9.621343631915807397375417258919, −8.910448690739695502978971615197, −8.028898592465782231154054654612, −7.09470047279332660367810644351, −6.50630169856512273080356922173, −5.41314369409222650548641678811, −4.59545733509252372833798654646, −3.38231779277823631748135116234, −2.57400704307847315821371629803, −0.980946675843082555455559878992, 1.05697479154615163147848532844, 2.31970741562293803614259658382, 3.73338923349036200463927854553, 4.34687651923312873619503153802, 5.65949449738204748271313028651, 6.16183749886901216457238240670, 7.23629684364117969840405342876, 8.265858805314398890881117614640, 8.668524257191216600906393757607, 9.622085922899466763630810063379

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