Properties

Label 2-2790-5.4-c1-0-29
Degree $2$
Conductor $2790$
Sign $0.894 + 0.447i$
Analytic cond. $22.2782$
Root an. cond. $4.71998$
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  i·2-s − 4-s + (1 − 2i)5-s + i·7-s + i·8-s + (−2 − i)10-s + 3·11-s + 4i·13-s + 14-s + 16-s + 19-s + (−1 + 2i)20-s − 3i·22-s + 5i·23-s + (−3 − 4i)25-s + 4·26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.447 − 0.894i)5-s + 0.377i·7-s + 0.353i·8-s + (−0.632 − 0.316i)10-s + 0.904·11-s + 1.10i·13-s + 0.267·14-s + 0.250·16-s + 0.229·19-s + (−0.223 + 0.447i)20-s − 0.639i·22-s + 1.04i·23-s + (−0.600 − 0.800i)25-s + 0.784·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2790\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 31\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(22.2782\)
Root analytic conductor: \(4.71998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2790} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2790,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.935445104\)
\(L(\frac12)\) \(\approx\) \(1.935445104\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 + (-1 + 2i)T \)
31 \( 1 + T \)
good7 \( 1 - iT - 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 - 5iT - 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 5iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 5T + 71T^{2} \)
73 \( 1 - 7iT - 73T^{2} \)
79 \( 1 + 3T + 79T^{2} \)
83 \( 1 + 2iT - 83T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−9.108393999182225739941639648002, −8.253766508185137760681900412881, −7.31056034392922677186730411044, −6.24903876178345709772211330298, −5.63780248374158553890614914115, −4.59690364956391631141625247499, −4.12131856928668054571521631682, −2.94097609371198186238399615557, −1.83752768207418797841057354295, −1.11379638749635992559441267076, 0.74370061332604406159554930944, 2.25213487854780927025181149392, 3.33136440373610905353862539951, 4.09133533823063571306132515451, 5.17423726607114961308056746940, 5.97114252062701652524117938244, 6.57578117983521645876269268085, 7.28234950793085691563411345893, 7.910757319965529520357772887768, 8.854691586096177477869225921079

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