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Properties

Label 2-810-15.14-c2-0-30
Degree $2$
Conductor $810$
Sign $0.984 + 0.176i$
Analytic cond. $22.0709$
Root an. cond. $4.69796$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + 2.00·4-s + (0.882 − 4.92i)5-s + 8.76i·7-s + 2.82·8-s + (1.24 − 6.96i)10-s − 13.4i·11-s + 17.2i·13-s + 12.3i·14-s + 4.00·16-s + 0.183·17-s + 34.7·19-s + (1.76 − 9.84i)20-s − 19.0i·22-s + 34.4·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.500·4-s + (0.176 − 0.984i)5-s + 1.25i·7-s + 0.353·8-s + (0.124 − 0.696i)10-s − 1.22i·11-s + 1.32i·13-s + 0.885i·14-s + 0.250·16-s + 0.0107·17-s + 1.82·19-s + (0.0882 − 0.492i)20-s − 0.864i·22-s + 1.49·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.984 + 0.176i$
Analytic conductor: \(22.0709\)
Root analytic conductor: \(4.69796\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1),\ 0.984 + 0.176i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.241096438\)
\(L(\frac12)\) \(\approx\) \(3.241096438\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
3 \( 1 \)
5 \( 1 + (-0.882 + 4.92i)T \)
good7 \( 1 - 8.76iT - 49T^{2} \)
11 \( 1 + 13.4iT - 121T^{2} \)
13 \( 1 - 17.2iT - 169T^{2} \)
17 \( 1 - 0.183T + 289T^{2} \)
19 \( 1 - 34.7T + 361T^{2} \)
23 \( 1 - 34.4T + 529T^{2} \)
29 \( 1 + 10.3iT - 841T^{2} \)
31 \( 1 - 20.6T + 961T^{2} \)
37 \( 1 + 8.75iT - 1.36e3T^{2} \)
41 \( 1 + 1.57iT - 1.68e3T^{2} \)
43 \( 1 + 40.5iT - 1.84e3T^{2} \)
47 \( 1 + 30.9T + 2.20e3T^{2} \)
53 \( 1 - 21.1T + 2.80e3T^{2} \)
59 \( 1 - 30.2iT - 3.48e3T^{2} \)
61 \( 1 - 72.6T + 3.72e3T^{2} \)
67 \( 1 - 6.55iT - 4.48e3T^{2} \)
71 \( 1 - 1.02iT - 5.04e3T^{2} \)
73 \( 1 + 16.8iT - 5.32e3T^{2} \)
79 \( 1 + 83.1T + 6.24e3T^{2} \)
83 \( 1 + 2.31T + 6.88e3T^{2} \)
89 \( 1 + 137. iT - 7.92e3T^{2} \)
97 \( 1 - 54.0iT - 9.40e3T^{2} \)
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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

−9.876133484011595243984332508184, −8.990642488760933012689516670831, −8.603645809075992055902364541586, −7.35859068815278446211082630768, −6.22196417777824288406357876057, −5.47011840683658640300856926927, −4.85692131564299497804589859532, −3.57923930630425526472624056942, −2.47879279325329528877187640124, −1.13372254428433567274590355554, 1.16328654115317578346600293617, 2.80351941715408432299205655842, 3.49582604855931174884517421938, 4.68833630999015052618069660326, 5.53445613620161706634195990844, 6.81805028569534191936294994012, 7.23452951978014473078263521817, 7.954833093407740138524636597017, 9.676985143361250563102877529829, 10.14123237503358373114471731079

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