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Properties

Label 2-6e4-9.7-c1-0-6
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: $\chi_{1296} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1296,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.822744864\)
\(L(\frac12)\) \(\approx\) \(1.822744864\)
\(L(\frac{3}{2})\) 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

−9.777757741237360002339327334269, −9.148643611844237141857628531640, −8.301573397976511241873904350283, −7.20919233048888620624394378167, −6.25727561079850022886066921031, −6.14468672509763761737082098870, −4.74319571739044989544841252986, −3.52973558185152318059097053045, −2.75555679847085373060118928734, −1.54328551351828958111255489604, 0.808117356047826066913450692885, 1.94029362432280859156290811229, 3.32954375323983113209664340505, 4.53568080039930419362873439029, 5.11651406377277662159806782209, 6.16009480359902246842373624528, 6.89098322128392487308365037656, 8.063439338348438485884420609619, 8.702492818396631087295748262906, 9.435276958998546136666193570814

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