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Properties

Label 2-1638-91.34-c1-0-16
Degree $2$
Conductor $1638$
Sign $0.596 - 0.802i$
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 + (1.15 − 1.15i)5-s + (2.02 + 1.70i)7-s + (0.707 − 0.707i)8-s − 1.63·10-s + (−1.37 + 1.37i)11-s + (−1.50 + 3.27i)13-s + (−0.222 − 2.63i)14-s − 1.00·16-s + 1.50·17-s + (−0.934 + 0.934i)19-s + (1.15 + 1.15i)20-s + 1.95·22-s + 4.19i·23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (0.517 − 0.517i)5-s + (0.763 + 0.645i)7-s + (0.250 − 0.250i)8-s − 0.517·10-s + (−0.415 + 0.415i)11-s + (−0.416 + 0.909i)13-s + (−0.0593 − 0.704i)14-s − 0.250·16-s + 0.365·17-s + (−0.214 + 0.214i)19-s + (0.258 + 0.258i)20-s + 0.415·22-s + 0.874i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.596 - 0.802i$
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: \((2,\ 1638,\ (\ :1/2),\ 0.596 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.239648168\)
\(L(\frac12)\) \(\approx\) \(1.239648168\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.02 - 1.70i)T \)
13 \( 1 + (1.50 - 3.27i)T \)
good5 \( 1 + (-1.15 + 1.15i)T - 5iT^{2} \)
11 \( 1 + (1.37 - 1.37i)T - 11iT^{2} \)
17 \( 1 - 1.50T + 17T^{2} \)
19 \( 1 + (0.934 - 0.934i)T - 19iT^{2} \)
23 \( 1 - 4.19iT - 23T^{2} \)
29 \( 1 - 0.406T + 29T^{2} \)
31 \( 1 + (2.31 - 2.31i)T - 31iT^{2} \)
37 \( 1 + (-0.257 + 0.257i)T - 37iT^{2} \)
41 \( 1 + (3.60 - 3.60i)T - 41iT^{2} \)
43 \( 1 - 2.46iT - 43T^{2} \)
47 \( 1 + (-1.41 - 1.41i)T + 47iT^{2} \)
53 \( 1 + 13.4T + 53T^{2} \)
59 \( 1 + (8.75 + 8.75i)T + 59iT^{2} \)
61 \( 1 + 11.7iT - 61T^{2} \)
67 \( 1 + (-11.1 - 11.1i)T + 67iT^{2} \)
71 \( 1 + (-5.18 - 5.18i)T + 71iT^{2} \)
73 \( 1 + (2.56 + 2.56i)T + 73iT^{2} \)
79 \( 1 - 1.42T + 79T^{2} \)
83 \( 1 + (4.90 - 4.90i)T - 83iT^{2} \)
89 \( 1 + (2.03 + 2.03i)T + 89iT^{2} \)
97 \( 1 + (-10.2 + 10.2i)T - 97iT^{2} \)
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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

−9.511904743208904978575214740976, −8.867521780670592628049911173631, −8.050839726042007192014503698040, −7.36360281946260851804601462524, −6.25476692157077443068638950029, −5.20593368290530001216991224010, −4.66402916591974278358478043277, −3.35748159278668978846734250887, −2.10996620570150460876557073760, −1.49134742088606634988512806481, 0.56107084863285174952276971209, 2.00903881593079046388107339821, 3.09408804758013918751245676679, 4.44279618628686517475575513802, 5.31208084467096922825653120197, 6.10249289319492360030927081327, 6.96482480927957836477027183804, 7.75895082680209195359061881618, 8.278477888912359674869518562120, 9.213125028468772481643832089221

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