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Properties

Label 2-1638-39.5-c1-0-24
Degree $2$
Conductor $1638$
Sign $-0.999 + 6.47e-5i$
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 + (−3.00 − 3.00i)5-s + (0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + 4.24i·10-s + (2.53 − 2.53i)11-s + (2.84 − 2.21i)13-s − 1.00i·14-s − 1.00·16-s − 7.16·17-s + (4.75 − 4.75i)19-s + (3.00 − 3.00i)20-s − 3.58·22-s + 5.70·23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−1.34 − 1.34i)5-s + (0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s + 1.34i·10-s + (0.764 − 0.764i)11-s + (0.789 − 0.614i)13-s − 0.267i·14-s − 0.250·16-s − 1.73·17-s + (1.09 − 1.09i)19-s + (0.671 − 0.671i)20-s − 0.764·22-s + 1.19·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 6.47e-5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 6.47e-5i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.999 + 6.47e-5i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.999 + 6.47e-5i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7715641164\)
\(L(\frac12)\) \(\approx\) \(0.7715641164\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
13 \( 1 + (-2.84 + 2.21i)T \)
good5 \( 1 + (3.00 + 3.00i)T + 5iT^{2} \)
11 \( 1 + (-2.53 + 2.53i)T - 11iT^{2} \)
17 \( 1 + 7.16T + 17T^{2} \)
19 \( 1 + (-4.75 + 4.75i)T - 19iT^{2} \)
23 \( 1 - 5.70T + 23T^{2} \)
29 \( 1 + 3.76iT - 29T^{2} \)
31 \( 1 + (-0.0835 + 0.0835i)T - 31iT^{2} \)
37 \( 1 + (1.86 + 1.86i)T + 37iT^{2} \)
41 \( 1 + (-3.47 - 3.47i)T + 41iT^{2} \)
43 \( 1 + 7.91iT - 43T^{2} \)
47 \( 1 + (-1.55 + 1.55i)T - 47iT^{2} \)
53 \( 1 + 5.22iT - 53T^{2} \)
59 \( 1 + (4.33 - 4.33i)T - 59iT^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + (-0.568 + 0.568i)T - 67iT^{2} \)
71 \( 1 + (-8.61 - 8.61i)T + 71iT^{2} \)
73 \( 1 + (3.94 + 3.94i)T + 73iT^{2} \)
79 \( 1 + 9.08T + 79T^{2} \)
83 \( 1 + (8.23 + 8.23i)T + 83iT^{2} \)
89 \( 1 + (8.69 - 8.69i)T - 89iT^{2} \)
97 \( 1 + (9.42 - 9.42i)T - 97iT^{2} \)
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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

−8.813154154118419061905338604877, −8.567579305129671570296814867961, −7.64918130482585391187718866210, −6.81970259339629045111716160468, −5.51688393225731214854779333105, −4.60995993261278464770365454710, −3.90309144909945061911658096648, −2.93174928961311721243690034804, −1.28481871482097173357654994138, −0.41228866206535740133007443464, 1.48541455408877624499990467511, 2.97141060849006173983771870954, 3.98683604049831958318621522597, 4.61712089163367173400740147386, 6.13557628331457380071410615119, 6.89897983418804214520335209121, 7.21900062594693097388310236414, 8.061756953058845987340822812256, 8.881707873925824893130121653816, 9.658647370008836971786101577147

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