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Properties

Label 2-6e4-36.23-c1-0-21
Degree $2$
Conductor $1296$
Sign $0.642 + 0.766i$
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  + (2.59 + 1.5i)5-s + (1.5 − 0.866i)7-s + (−2.59 − 4.5i)11-s + (1 − 1.73i)13-s − 6i·17-s − 6.92i·19-s + (2 + 3.46i)25-s + (−5.19 + 3i)29-s + (−4.5 − 2.59i)31-s + 5.19·35-s + 8·37-s + (9 − 5.19i)43-s + (5.19 + 9i)47-s + (−2 + 3.46i)49-s + 9i·53-s + ⋯
L(s)  = 1  + (1.16 + 0.670i)5-s + (0.566 − 0.327i)7-s + (−0.783 − 1.35i)11-s + (0.277 − 0.480i)13-s − 1.45i·17-s − 1.58i·19-s + (0.400 + 0.692i)25-s + (−0.964 + 0.557i)29-s + (−0.808 − 0.466i)31-s + 0.878·35-s + 1.31·37-s + (1.37 − 0.792i)43-s + (0.757 + 1.31i)47-s + (−0.285 + 0.494i)49-s + 1.23i·53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(10.3486\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (863, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.994080772\)
\(L(\frac12)\) \(\approx\) \(1.994080772\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 - 1.5i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.5 + 0.866i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.59 + 4.5i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 6.92iT - 19T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (5.19 - 3i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.5 + 2.59i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-9 + 5.19i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.19 - 9i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + (5.19 - 9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3 + 1.73i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (-3 + 1.73i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.59 - 4.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.363587045346512442658567819164, −9.018585815358247672465948666252, −7.72253084523371461803828688607, −7.21130812373886920530098881443, −5.99233258507094336906712548710, −5.57209092657680689699535401521, −4.52438560038646191763743599519, −3.02325865326981425920612102553, −2.48232884004690307500733392751, −0.837075201525958871007728023307, 1.73932551197455265844130717145, 2.02752263734196660047599907058, 3.83465941898381999698032927350, 4.78754293319957480242698568406, 5.63870512433124292640436660338, 6.17863368502828236627710621693, 7.47845702382015483374178690110, 8.177753286158167706664971688816, 9.028596266009866566594451705895, 9.818941360434457394457440965614

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