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Properties

Label 2-6e4-9.7-c1-0-3
Degree $2$
Conductor $1296$
Sign $-0.766 - 0.642i$
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 + 3.46i)5-s + (−1.5 + 2.59i)7-s + (−2 + 3.46i)11-s + (−0.5 − 0.866i)13-s + 4·17-s + 19-s + (−2 − 3.46i)23-s + (−5.49 + 9.52i)25-s + (−2 − 3.46i)31-s − 12·35-s − 9·37-s + (−4 + 6.92i)43-s + (6 − 10.3i)47-s + (−1 − 1.73i)49-s + 8·53-s + ⋯
L(s)  = 1  + (0.894 + 1.54i)5-s + (−0.566 + 0.981i)7-s + (−0.603 + 1.04i)11-s + (−0.138 − 0.240i)13-s + 0.970·17-s + 0.229·19-s + (−0.417 − 0.722i)23-s + (−1.09 + 1.90i)25-s + (−0.359 − 0.622i)31-s − 2.02·35-s − 1.47·37-s + (−0.609 + 1.05i)43-s + (0.875 − 1.51i)47-s + (−0.142 − 0.247i)49-s + 1.09·53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $-0.766 - 0.642i$
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: \((2,\ 1296,\ (\ :1/2),\ -0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.456834492\)
\(L(\frac12)\) \(\approx\) \(1.456834492\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 9T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8T + 53T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4 - 6.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
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

−10.02170884125391701651409685082, −9.480909892471938857558869721817, −8.332802935702372913146679334186, −7.29505925695317916880085097214, −6.71989860919864914066355616211, −5.80859836593850024484642033298, −5.24122196836954494326969907862, −3.62855663242470304394979906726, −2.67216340198268497461872986171, −2.09039873743732264497871344375, 0.58908120434667112569858788197, 1.64762828548782518679932198388, 3.21015025661757063488451733208, 4.18629914470806684932915321082, 5.36906978344460176773306938762, 5.66371053168327754620653049130, 6.86744122284269292709356725206, 7.83627492024691991556789911610, 8.638132438245300380460439515168, 9.333154307785100565864704958112

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