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Properties

Label 2-1638-91.34-c1-0-33
Degree $2$
Conductor $1638$
Sign $0.555 + 0.831i$
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 + (−2.16 + 2.16i)5-s + (−0.292 − 2.62i)7-s + (−0.707 + 0.707i)8-s − 3.06·10-s + (−0.516 + 0.516i)11-s + (−3.60 − 0.0469i)13-s + (1.65 − 2.06i)14-s − 1.00·16-s − 2.57·17-s + (3.82 − 3.82i)19-s + (−2.16 − 2.16i)20-s − 0.729·22-s − 6.97i·23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.969 + 0.969i)5-s + (−0.110 − 0.993i)7-s + (−0.250 + 0.250i)8-s − 0.969·10-s + (−0.155 + 0.155i)11-s + (−0.999 − 0.0130i)13-s + (0.441 − 0.552i)14-s − 0.250·16-s − 0.624·17-s + (0.876 − 0.876i)19-s + (−0.484 − 0.484i)20-s − 0.155·22-s − 1.45i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.555 + 0.831i$
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: \((2,\ 1638,\ (\ :1/2),\ 0.555 + 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9342455289\)
\(L(\frac12)\) \(\approx\) \(0.9342455289\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.292 + 2.62i)T \)
13 \( 1 + (3.60 + 0.0469i)T \)
good5 \( 1 + (2.16 - 2.16i)T - 5iT^{2} \)
11 \( 1 + (0.516 - 0.516i)T - 11iT^{2} \)
17 \( 1 + 2.57T + 17T^{2} \)
19 \( 1 + (-3.82 + 3.82i)T - 19iT^{2} \)
23 \( 1 + 6.97iT - 23T^{2} \)
29 \( 1 - 6.32T + 29T^{2} \)
31 \( 1 + (-4.33 + 4.33i)T - 31iT^{2} \)
37 \( 1 + (3.58 - 3.58i)T - 37iT^{2} \)
41 \( 1 + (0.176 - 0.176i)T - 41iT^{2} \)
43 \( 1 + 7.89iT - 43T^{2} \)
47 \( 1 + (-1.41 - 1.41i)T + 47iT^{2} \)
53 \( 1 - 0.514T + 53T^{2} \)
59 \( 1 + (3.17 + 3.17i)T + 59iT^{2} \)
61 \( 1 - 6.41iT - 61T^{2} \)
67 \( 1 + (3.96 + 3.96i)T + 67iT^{2} \)
71 \( 1 + (8.12 + 8.12i)T + 71iT^{2} \)
73 \( 1 + (11.6 + 11.6i)T + 73iT^{2} \)
79 \( 1 - 10.9T + 79T^{2} \)
83 \( 1 + (-1.01 + 1.01i)T - 83iT^{2} \)
89 \( 1 + (-1.10 - 1.10i)T + 89iT^{2} \)
97 \( 1 + (-9.34 + 9.34i)T - 97iT^{2} \)
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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

−9.163905521373811315974409379594, −8.121767033425612435649943046333, −7.43089954755807465780637349619, −6.94958555816135842383035782454, −6.31795082787944279429720473387, −4.81523713864628385956656816567, −4.40511278259720896243229101254, −3.31054835219336672642433181524, −2.57598628236058992767040696070, −0.31718871934922188188813968764, 1.30118732776540395133974524502, 2.63026418181702397865238868743, 3.54838909172198946798742487282, 4.60653847130102718567520180598, 5.16803580182237123921759878352, 5.99013835434365097262005422126, 7.18431751748238828999145323248, 8.047403873089596779088089825968, 8.745731491439918870757651875727, 9.505942485468264620613871652849

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