# Properties

 Label 2-6e4-9.7-c1-0-21 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 $1$

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## 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: $$1$$ 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 + (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

−9.099326134506888976388389403865, −8.485723398831119828084220288601, −7.79823939270221944414961945552, −6.58988464463012920252897554945, −5.67241571895452984371100486874, −4.94500255728065287148824078363, −3.95468530329711412549310601519, −3.02658042727998162700121981244, −1.40099024325289169355249570498, 0, 2.07877301648297611464705464658, 3.34050857360392175056115680779, 3.90451297928513085658154998130, 4.87336213796623534854101082262, 6.63157983898903221583972715568, 6.80814427043802261992235946776, 7.36656014966167808402898350612, 8.445952931243158883147403713774, 9.500649251207420568219218988039