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Properties

Label 2-810-135.113-c1-0-16
Degree $2$
Conductor $810$
Sign $0.174 + 0.984i$
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  + (0.996 + 0.0871i)2-s + (0.984 + 0.173i)4-s + (1.30 − 1.81i)5-s + (−3.63 − 2.54i)7-s + (0.965 + 0.258i)8-s + (1.46 − 1.69i)10-s + (−1.06 + 2.91i)11-s + (−0.443 − 5.06i)13-s + (−3.39 − 2.85i)14-s + (0.939 + 0.342i)16-s + (2.90 − 0.778i)17-s + (4.83 − 2.78i)19-s + (1.60 − 1.55i)20-s + (−1.31 + 2.81i)22-s + (−3.26 − 4.66i)23-s + ⋯
L(s)  = 1  + (0.704 + 0.0616i)2-s + (0.492 + 0.0868i)4-s + (0.585 − 0.810i)5-s + (−1.37 − 0.961i)7-s + (0.341 + 0.0915i)8-s + (0.462 − 0.534i)10-s + (−0.319 + 0.879i)11-s + (−0.122 − 1.40i)13-s + (−0.908 − 0.762i)14-s + (0.234 + 0.0855i)16-s + (0.704 − 0.188i)17-s + (1.10 − 0.639i)19-s + (0.358 − 0.348i)20-s + (−0.279 + 0.599i)22-s + (−0.681 − 0.973i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.174 + 0.984i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (773, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ 0.174 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61278 - 1.35249i\)
\(L(\frac12)\) \(\approx\) \(1.61278 - 1.35249i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.996 - 0.0871i)T \)
3 \( 1 \)
5 \( 1 + (-1.30 + 1.81i)T \)
good7 \( 1 + (3.63 + 2.54i)T + (2.39 + 6.57i)T^{2} \)
11 \( 1 + (1.06 - 2.91i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (0.443 + 5.06i)T + (-12.8 + 2.25i)T^{2} \)
17 \( 1 + (-2.90 + 0.778i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-4.83 + 2.78i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.26 + 4.66i)T + (-7.86 + 21.6i)T^{2} \)
29 \( 1 + (-0.945 + 0.793i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.863 - 4.89i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (0.154 + 0.578i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.230 - 0.275i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (-0.843 - 1.80i)T + (-27.6 + 32.9i)T^{2} \)
47 \( 1 + (4.62 - 6.60i)T + (-16.0 - 44.1i)T^{2} \)
53 \( 1 + (-5.61 + 5.61i)T - 53iT^{2} \)
59 \( 1 + (-4.22 + 1.53i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.293 - 1.66i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-8.65 + 0.757i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (4.54 + 2.62i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-0.146 + 0.545i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.05 - 4.82i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (1.20 - 13.7i)T + (-81.7 - 14.4i)T^{2} \)
89 \( 1 + (-6.98 - 12.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.7 + 6.41i)T + (62.3 - 74.3i)T^{2} \)
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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

−9.980031589903715458123911252187, −9.589678255907138238449248036544, −8.195582325601890927100610138156, −7.32486812740935975755534635768, −6.49233628871227005733442779331, −5.50582539447285976824279663271, −4.76897720490499659310119147651, −3.59587593432287282745721292376, −2.63320959817974818903142262011, −0.824391165709954048282918423034, 2.00854879813881670461854020719, 3.08707480540652650007382862053, 3.71889575529408300698024933799, 5.47861405279291805305706940677, 5.93000689908664546650172427586, 6.68298804379991313077913619245, 7.62760584726849382511812180320, 8.991314997414110766737951033390, 9.727004680652257653863722732780, 10.31247623558029121658997075101

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