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Properties

Label 2-810-15.14-c2-0-37
Degree $2$
Conductor $810$
Sign $-0.701 + 0.712i$
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 + (3.56 + 3.50i)5-s + 3.17i·7-s − 2.82·8-s + (−5.03 − 4.96i)10-s − 15.6i·11-s − 1.33i·13-s − 4.49i·14-s + 4.00·16-s − 23.4·17-s − 23.3·19-s + (7.12 + 7.01i)20-s + 22.1i·22-s − 28.6·23-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.500·4-s + (0.712 + 0.701i)5-s + 0.454i·7-s − 0.353·8-s + (−0.503 − 0.496i)10-s − 1.42i·11-s − 0.102i·13-s − 0.321i·14-s + 0.250·16-s − 1.38·17-s − 1.22·19-s + (0.356 + 0.350i)20-s + 1.00i·22-s − 1.24·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $-0.701 + 0.712i$
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: \((2,\ 810,\ (\ :1),\ -0.701 + 0.712i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3594649206\)
\(L(\frac12)\) \(\approx\) \(0.3594649206\)
\(L(2)\) 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 \( 1 + (-3.56 - 3.50i)T \)
good7 \( 1 - 3.17iT - 49T^{2} \)
11 \( 1 + 15.6iT - 121T^{2} \)
13 \( 1 + 1.33iT - 169T^{2} \)
17 \( 1 + 23.4T + 289T^{2} \)
19 \( 1 + 23.3T + 361T^{2} \)
23 \( 1 + 28.6T + 529T^{2} \)
29 \( 1 + 33.1iT - 841T^{2} \)
31 \( 1 - 18.5T + 961T^{2} \)
37 \( 1 + 3.95iT - 1.36e3T^{2} \)
41 \( 1 + 60.3iT - 1.68e3T^{2} \)
43 \( 1 + 2.83iT - 1.84e3T^{2} \)
47 \( 1 + 71.6T + 2.20e3T^{2} \)
53 \( 1 - 14.3T + 2.80e3T^{2} \)
59 \( 1 - 72.3iT - 3.48e3T^{2} \)
61 \( 1 - 11.2T + 3.72e3T^{2} \)
67 \( 1 - 3.35iT - 4.48e3T^{2} \)
71 \( 1 + 40.0iT - 5.04e3T^{2} \)
73 \( 1 - 86.6iT - 5.32e3T^{2} \)
79 \( 1 + 90.5T + 6.24e3T^{2} \)
83 \( 1 + 102.T + 6.88e3T^{2} \)
89 \( 1 + 144. iT - 7.92e3T^{2} \)
97 \( 1 + 74.8iT - 9.40e3T^{2} \)
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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

−9.787376547060781517945329638210, −8.742759322375842865784886686826, −8.365958599828181698461661366021, −7.08893362262391313930775026069, −6.18613835872597763430627983901, −5.76978085233843572816691411342, −4.11266284471223625499359913394, −2.77697886503230922270707887355, −1.97767213597914056541948157793, −0.14048877331465481801848233553, 1.55191828464483824012875652012, 2.35507227423974783122872826148, 4.19662372050026269600386717373, 4.91386384610666658227436076024, 6.30966319922120445311215474951, 6.84855167245432390484117460428, 8.010092934861869049971862743560, 8.725675225210106168608846407648, 9.603032063392164653458028617623, 10.13411098515994637686185371395

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