Properties

Label 2-6e4-3.2-c4-0-76
Degree $2$
Conductor $1296$
Sign $i$
Analytic cond. $133.967$
Root an. cond. $11.5744$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 12.2i·5-s + 14.2·7-s − 104. i·11-s + 75.2·13-s − 341. i·17-s + 706.·19-s − 596. i·23-s + 474.·25-s + 1.30e3i·29-s − 1.02e3·31-s − 175. i·35-s + 563.·37-s − 99.1i·41-s + 896.·43-s − 430. i·47-s + ⋯
L(s)  = 1  − 0.491i·5-s + 0.291·7-s − 0.860i·11-s + 0.445·13-s − 1.18i·17-s + 1.95·19-s − 1.12i·23-s + 0.758·25-s + 1.54i·29-s − 1.07·31-s − 0.143i·35-s + 0.411·37-s − 0.0589i·41-s + 0.484·43-s − 0.194i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $i$
Analytic conductor: \(133.967\)
Root analytic conductor: \(11.5744\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :2),\ i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.421156482\)
\(L(\frac12)\) \(\approx\) \(2.421156482\)
\(L(3)\) 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 \( 1 + 12.2iT - 625T^{2} \)
7 \( 1 - 14.2T + 2.40e3T^{2} \)
11 \( 1 + 104. iT - 1.46e4T^{2} \)
13 \( 1 - 75.2T + 2.85e4T^{2} \)
17 \( 1 + 341. iT - 8.35e4T^{2} \)
19 \( 1 - 706.T + 1.30e5T^{2} \)
23 \( 1 + 596. iT - 2.79e5T^{2} \)
29 \( 1 - 1.30e3iT - 7.07e5T^{2} \)
31 \( 1 + 1.02e3T + 9.23e5T^{2} \)
37 \( 1 - 563.T + 1.87e6T^{2} \)
41 \( 1 + 99.1iT - 2.82e6T^{2} \)
43 \( 1 - 896.T + 3.41e6T^{2} \)
47 \( 1 + 430. iT - 4.87e6T^{2} \)
53 \( 1 - 5.27e3iT - 7.89e6T^{2} \)
59 \( 1 + 5.63e3iT - 1.21e7T^{2} \)
61 \( 1 - 1.13e3T + 1.38e7T^{2} \)
67 \( 1 - 1.35e3T + 2.01e7T^{2} \)
71 \( 1 + 5.68e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.23e3T + 2.83e7T^{2} \)
79 \( 1 - 6.13e3T + 3.89e7T^{2} \)
83 \( 1 + 7.50e3iT - 4.74e7T^{2} \)
89 \( 1 - 8.72e3iT - 6.27e7T^{2} \)
97 \( 1 - 5.44e3T + 8.85e7T^{2} \)
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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

−9.049747521394862466781773457512, −8.119574759650668694513473590671, −7.34693958794132735436081983891, −6.43793444177363218725212014305, −5.32171383978194777182165963671, −4.91061062405111062542685699737, −3.57581662478201854313184469686, −2.79804712779173639960879011496, −1.31522420639132507973579204862, −0.56209821737177858137407758456, 1.08293445756007305172834738861, 2.06043606666157125440888431493, 3.25727928781784306332830589183, 4.06963931122092026385245872032, 5.19766032021286874244052450540, 5.94619851986537433857302650238, 6.98661541683233997335892136963, 7.61947229703703482544866363001, 8.394951446901511007236113552636, 9.542194211542448743804626459258

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