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Properties

Label 2-6e4-9.7-c1-0-22
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 − 2.59i)5-s + (1 − 1.73i)7-s + (−3 + 5.19i)11-s + (−2.5 − 4.33i)13-s − 3·17-s − 2·19-s + (3 + 5.19i)23-s + (−2 + 3.46i)25-s + (−1.5 + 2.59i)29-s + (−2 − 3.46i)31-s − 6·35-s + 5·37-s + (3 + 5.19i)41-s + (−5 + 8.66i)43-s + (1.50 + 2.59i)49-s + ⋯
L(s)  = 1  + (−0.670 − 1.16i)5-s + (0.377 − 0.654i)7-s + (−0.904 + 1.56i)11-s + (−0.693 − 1.20i)13-s − 0.727·17-s − 0.458·19-s + (0.625 + 1.08i)23-s + (−0.400 + 0.692i)25-s + (−0.278 + 0.482i)29-s + (−0.359 − 0.622i)31-s − 1.01·35-s + 0.821·37-s + (0.468 + 0.811i)41-s + (−0.762 + 1.32i)43-s + (0.214 + 0.371i)49-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: \(1\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ -0.766 - 0.642i)\)

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 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.5 + 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5 - 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + T + 73T^{2} \)
79 \( 1 + (5 - 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
97 \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.259650295228427430645924809421, −7.970110469822135180309382130496, −7.84021219639205210408512312379, −6.95530601917106787739119657215, −5.50199424166099694755149716467, −4.72026964119258487434264193119, −4.30982852181024666686001526412, −2.84601146801517810354973447351, −1.47282272623686094791119897549, 0, 2.25622874780795178387995546928, 2.97092318778329899567688534320, 4.07300718772695014351583245664, 5.10290378074997383625600015200, 6.14522234673339724413771065041, 6.87275115150276690653262973933, 7.68876969644185879207134511653, 8.577333685401399343234855340031, 9.113352002665881162914952293930

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