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Properties

Label 2-6e4-9.4-c1-0-13
Degree $2$
Conductor $1296$
Sign $0.173 + 0.984i$
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  + (−1.5 + 2.59i)5-s + (−0.5 − 0.866i)7-s + (−1.5 − 2.59i)11-s + (2 − 3.46i)13-s − 2·19-s + (−3 + 5.19i)23-s + (−2 − 3.46i)25-s + (−3 − 5.19i)29-s + (2.5 − 4.33i)31-s + 3·35-s + 2·37-s + (3 − 5.19i)41-s + (−5 − 8.66i)43-s + (3 + 5.19i)47-s + (3 − 5.19i)49-s + ⋯
L(s)  = 1  + (−0.670 + 1.16i)5-s + (−0.188 − 0.327i)7-s + (−0.452 − 0.783i)11-s + (0.554 − 0.960i)13-s − 0.458·19-s + (−0.625 + 1.08i)23-s + (−0.400 − 0.692i)25-s + (−0.557 − 0.964i)29-s + (0.449 − 0.777i)31-s + 0.507·35-s + 0.328·37-s + (0.468 − 0.811i)41-s + (−0.762 − 1.32i)43-s + (0.437 + 0.757i)47-s + (0.428 − 0.742i)49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1296\)    =    \(2^{4} \cdot 3^{4}\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(10.3486\)
Root analytic conductor: \(3.21692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1296} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ 0.173 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8924581009\)
\(L(\frac12)\) \(\approx\) \(0.8924581009\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
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 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7 + 12.1i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 7T + 73T^{2} \)
79 \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.5 + 2.59i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 18T + 89T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.656006873732270803153995633813, −8.413971585554655254693153641808, −7.83609555994059738411538301002, −7.11695985635625060933175497586, −6.15837862759726484282065971427, −5.46345401361076513034899102011, −3.92096703262733784497332058272, −3.44322270379388536072764451070, −2.36182743128603628280398755613, −0.39594265362025642370081711560, 1.30144393276342706793242850071, 2.60322991193936342984643976652, 4.09898884746470333322458933721, 4.54736471386251336464197687542, 5.55335235845374044294954377329, 6.58559162243052681813754138401, 7.44804446916367992049576119955, 8.521399002859435344255609083733, 8.740725989049939507801918146223, 9.746266926549570975963666163121

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