Properties

Label 2-1638-13.12-c1-0-19
Degree $2$
Conductor $1638$
Sign $0.554 - 0.832i$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
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 + 2i·5-s i·7-s i·8-s − 2·10-s + (2 − 3i)13-s + 14-s + 16-s + 2·17-s − 4i·19-s − 2i·20-s + 6·23-s + 25-s + (3 + 2i)26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.894i·5-s − 0.377i·7-s − 0.353i·8-s − 0.632·10-s + (0.554 − 0.832i)13-s + 0.267·14-s + 0.250·16-s + 0.485·17-s − 0.917i·19-s − 0.447i·20-s + 1.25·23-s + 0.200·25-s + (0.588 + 0.392i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.554 - 0.832i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.554 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.690956669\)
\(L(\frac12)\) \(\approx\) \(1.690956669\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + iT \)
13 \( 1 + (-2 + 3i)T \)
good5 \( 1 - 2iT - 5T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 14iT - 83T^{2} \)
89 \( 1 - 4iT - 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−9.424698992451957286159519272060, −8.595071690056888259044084140301, −7.75638654433286324095934684822, −7.06286264566425336447786157760, −6.45353718205156970366882323021, −5.52967677766559342397443899790, −4.66830019815387131361063796642, −3.50908506190115822744657955592, −2.76494907188270627237837438418, −0.924820628258119128540482393297, 0.977408154245149686882164570202, 1.97509705869079714685717712568, 3.24666959081167690420024510323, 4.16199777834455602780680036225, 5.04112035904256504823063621828, 5.77219638747517919045293616973, 6.84202525551738398982228498201, 7.934325375544616487882788847408, 8.749168164747724714042925181899, 9.152418986267061821224835424456

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