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Properties

Label 2-810-9.7-c1-0-3
Degree $2$
Conductor $810$
Sign $0.173 - 0.984i$
Analytic cond. $6.46788$
Root an. cond. $2.54320$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−1 + 1.73i)7-s − 0.999·8-s − 0.999·10-s + (−3 + 5.19i)11-s + (2 + 3.46i)13-s + (0.999 + 1.73i)14-s + (−0.5 + 0.866i)16-s − 6·17-s − 4·19-s + (−0.499 + 0.866i)20-s + (3 + 5.19i)22-s + (−0.499 + 0.866i)25-s + 3.99·26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.223 − 0.387i)5-s + (−0.377 + 0.654i)7-s − 0.353·8-s − 0.316·10-s + (−0.904 + 1.56i)11-s + (0.554 + 0.960i)13-s + (0.267 + 0.462i)14-s + (−0.125 + 0.216i)16-s − 1.45·17-s − 0.917·19-s + (−0.111 + 0.193i)20-s + (0.639 + 1.10i)22-s + (−0.0999 + 0.173i)25-s + 0.784·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.173 - 0.984i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.628139 + 0.527071i\)
\(L(\frac12)\) \(\approx\) \(0.628139 + 0.527071i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4 - 6.92i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (3 + 5.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.47971562234997681470438693717, −9.612177295253383939311759530411, −8.947470716530354110477846448794, −8.042113016837210658277325359435, −6.80134109538508459079806331620, −6.03857788877013046871244035435, −4.61645995469969852790622004617, −4.38993071773018760611484621593, −2.72105084104399368851911476777, −1.87107147946259653933931644430, 0.34198757622228646662232513751, 2.75808607643885212279123698217, 3.61171609615017146047630509246, 4.66655724417253610145915158142, 5.87517717497737689349266025337, 6.43996645843552131653350404093, 7.44881700850522627078526022028, 8.304652717885101175682476443910, 8.864964420080332731356074613770, 10.34589126991547620014561408667

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