Properties

 Label 2-6e4-36.7-c0-0-3 Degree $2$ Conductor $1296$ Sign $-0.173 + 0.984i$ Analytic cond. $0.646788$ Root an. cond. $0.804231$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (−1.5 − 0.866i)7-s + (−0.5 − 0.866i)13-s − 1.73i·19-s + (0.5 − 0.866i)25-s − 37-s + (1 + 1.73i)49-s + (0.5 − 0.866i)61-s + (−1.5 + 0.866i)67-s + 73-s + (1.5 + 0.866i)79-s + 1.73i·91-s + (−0.5 + 0.866i)97-s + (1.5 − 0.866i)103-s − 2·109-s + ⋯
 L(s)  = 1 + (−1.5 − 0.866i)7-s + (−0.5 − 0.866i)13-s − 1.73i·19-s + (0.5 − 0.866i)25-s − 37-s + (1 + 1.73i)49-s + (0.5 − 0.866i)61-s + (−1.5 + 0.866i)67-s + 73-s + (1.5 + 0.866i)79-s + 1.73i·91-s + (−0.5 + 0.866i)97-s + (1.5 − 0.866i)103-s − 2·109-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, 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: $$0.646788$$ Root analytic conductor: $$0.804231$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{1296} (703, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1296,\ (\ :0),\ -0.173 + 0.984i)$$

Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6997494220$$ $$L(\frac12)$$ $$\approx$$ $$0.6997494220$$ $$L(1)$$ 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 + (-0.5 + 0.866i)T^{2}$$
7 $$1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}$$
11 $$1 + (0.5 + 0.866i)T^{2}$$
13 $$1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}$$
17 $$1 + T^{2}$$
19 $$1 + 1.73iT - T^{2}$$
23 $$1 + (0.5 - 0.866i)T^{2}$$
29 $$1 + (-0.5 - 0.866i)T^{2}$$
31 $$1 + (0.5 - 0.866i)T^{2}$$
37 $$1 + T + T^{2}$$
41 $$1 + (-0.5 + 0.866i)T^{2}$$
43 $$1 + (0.5 + 0.866i)T^{2}$$
47 $$1 + (0.5 + 0.866i)T^{2}$$
53 $$1 + T^{2}$$
59 $$1 + (0.5 - 0.866i)T^{2}$$
61 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
67 $$1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}$$
71 $$1 - T^{2}$$
73 $$1 - T + T^{2}$$
79 $$1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2}$$
83 $$1 + (0.5 + 0.866i)T^{2}$$
89 $$1 + T^{2}$$
97 $$1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)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

−9.674812035895073043306014984345, −8.977847681960974324485170616804, −7.930610108934299194122012390095, −6.95689306953257375087321554908, −6.60041019891404903192229002763, −5.42565465965336258914538845624, −4.44429522553574789214401890130, −3.37506936342690733407412296345, −2.61472737352510754164452097648, −0.57489603736880578017237761974, 1.89411824156457209905851467830, 3.06804887972638879037669248962, 3.86408416361333946066487271286, 5.16697932780935952388625824056, 6.01671241690362663200240029680, 6.67904247366919260201388449221, 7.56187268558252173153700921643, 8.669060831731531756725945666178, 9.311455163150950780164805280022, 9.919836076926915501974594132398