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Properties

Label 2-6e4-36.11-c1-0-21
Degree $2$
Conductor $1296$
Sign $-0.984 - 0.173i$
Analytic cond. $10.3486$
Root an. cond. $3.21692$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3 − 1.73i)7-s + (1 + 1.73i)13-s − 3.46i·19-s + (−2.5 + 4.33i)25-s + (−9 + 5.19i)31-s − 10·37-s + (−9 − 5.19i)43-s + (2.5 + 4.33i)49-s + (−7 + 12.1i)61-s + (−3 + 1.73i)67-s + 10·73-s + (−15 − 8.66i)79-s − 6.92i·91-s + (7 − 12.1i)97-s + (3 − 1.73i)103-s + ⋯
L(s)  = 1  + (−1.13 − 0.654i)7-s + (0.277 + 0.480i)13-s − 0.794i·19-s + (−0.5 + 0.866i)25-s + (−1.61 + 0.933i)31-s − 1.64·37-s + (−1.37 − 0.792i)43-s + (0.357 + 0.618i)49-s + (−0.896 + 1.55i)61-s + (−0.366 + 0.211i)67-s + 1.17·73-s + (−1.68 − 0.974i)79-s − 0.726i·91-s + (0.710 − 1.23i)97-s + (0.295 − 0.170i)103-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $-0.984 - 0.173i$
Analytic conductor: \(10.3486\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ -0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (3 + 1.73i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (9 - 5.19i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (9 + 5.19i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7 - 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3 - 1.73i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10T + 73T^{2} \)
79 \( 1 + (15 + 8.66i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-7 + 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.188316340027022761753494398617, −8.641213587679830770238799668031, −7.22912606979012288182343886105, −6.99434414576939500729354623063, −5.95146928908529483340917740141, −4.98741009225146465774408250848, −3.81266482312164629058496131195, −3.18437635711038415281721514250, −1.68501140599519423408150140875, 0, 1.90122466674381610432524343009, 3.11435795830023942203295875344, 3.85258078133306121299589384489, 5.20509341476204425358857392739, 5.98953758786361121439203569905, 6.64067407637014960138314739604, 7.71455632940894528481633771734, 8.500749533110170567923410202697, 9.361937964888382320503327288224

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