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Properties

Label 2-1638-91.34-c1-0-8
Degree $2$
Conductor $1638$
Sign $-0.898 - 0.439i$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.0951 + 0.0951i)5-s + (−2.64 + 0.0951i)7-s + (−0.707 + 0.707i)8-s − 0.134·10-s + (3.64 − 3.64i)11-s + (2 + 3i)13-s + (−1.93 − 1.80i)14-s − 1.00·16-s − 5.98·17-s + (−4.19 + 4.19i)19-s + (−0.0951 − 0.0951i)20-s + 5.15·22-s + 4.69i·23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.0425 + 0.0425i)5-s + (−0.999 + 0.0359i)7-s + (−0.250 + 0.250i)8-s − 0.0425·10-s + (1.09 − 1.09i)11-s + (0.554 + 0.832i)13-s + (−0.517 − 0.481i)14-s − 0.250·16-s − 1.45·17-s + (−0.961 + 0.961i)19-s + (−0.0212 − 0.0212i)20-s + 1.09·22-s + 0.978i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.898 - 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.898 - 0.439i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.898 - 0.439i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.243547130\)
\(L(\frac12)\) \(\approx\) \(1.243547130\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
7 \( 1 + (2.64 - 0.0951i)T \)
13 \( 1 + (-2 - 3i)T \)
good5 \( 1 + (0.0951 - 0.0951i)T - 5iT^{2} \)
11 \( 1 + (-3.64 + 3.64i)T - 11iT^{2} \)
17 \( 1 + 5.98T + 17T^{2} \)
19 \( 1 + (4.19 - 4.19i)T - 19iT^{2} \)
23 \( 1 - 4.69iT - 23T^{2} \)
29 \( 1 - 4.59T + 29T^{2} \)
31 \( 1 + (0.739 - 0.739i)T - 31iT^{2} \)
37 \( 1 + (4.83 - 4.83i)T - 37iT^{2} \)
41 \( 1 + (3.04 - 3.04i)T - 41iT^{2} \)
43 \( 1 - 8.78iT - 43T^{2} \)
47 \( 1 + (3.28 + 3.28i)T + 47iT^{2} \)
53 \( 1 + 1.09T + 53T^{2} \)
59 \( 1 + (1.30 + 1.30i)T + 59iT^{2} \)
61 \( 1 + 5.98iT - 61T^{2} \)
67 \( 1 + (0.0454 + 0.0454i)T + 67iT^{2} \)
71 \( 1 + (8.69 + 8.69i)T + 71iT^{2} \)
73 \( 1 + (4.83 + 4.83i)T + 73iT^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 + (-1.28 + 1.28i)T - 83iT^{2} \)
89 \( 1 + (-9.21 - 9.21i)T + 89iT^{2} \)
97 \( 1 + (9.97 - 9.97i)T - 97iT^{2} \)
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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

−9.433940258902783044080195104008, −8.890491327300257356537392109982, −8.225103381465315958527307051613, −6.97577636386588463807134095794, −6.38650249159215941214972635404, −6.01477057691556299474267395241, −4.69270940200859854654629592158, −3.77324563450248729895184072517, −3.20813367552274522795138924443, −1.64395865161756890935585581031, 0.38670534097294451385531708944, 2.02758102883215401352597487976, 2.91832400374223875793323589301, 4.13356340869101252164952771456, 4.50669208104873697897275256970, 5.84206024722998312525370334690, 6.67006295925041939026392324121, 7.01613930221356753784434487288, 8.690397745701975238916644083081, 8.917117708053317096667714580346

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