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Properties

Label 2-1638-39.5-c1-0-26
Degree $2$
Conductor $1638$
Sign $0.0413 + 0.999i$
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 + (−1.40 − 1.40i)5-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s − 1.98i·10-s + (−0.369 + 0.369i)11-s + (3.37 − 1.27i)13-s − 1.00i·14-s − 1.00·16-s − 0.875·17-s + (−2.06 + 2.06i)19-s + (1.40 − 1.40i)20-s − 0.522·22-s − 5.51·23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (−0.626 − 0.626i)5-s + (−0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s − 0.626i·10-s + (−0.111 + 0.111i)11-s + (0.934 − 0.354i)13-s − 0.267i·14-s − 0.250·16-s − 0.212·17-s + (−0.474 + 0.474i)19-s + (0.313 − 0.313i)20-s − 0.111·22-s − 1.15·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0413 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0413 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.0413 + 0.999i$
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: \((2,\ 1638,\ (\ :1/2),\ 0.0413 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.080711712\)
\(L(\frac12)\) \(\approx\) \(1.080711712\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.37 + 1.27i)T \)
good5 \( 1 + (1.40 + 1.40i)T + 5iT^{2} \)
11 \( 1 + (0.369 - 0.369i)T - 11iT^{2} \)
17 \( 1 + 0.875T + 17T^{2} \)
19 \( 1 + (2.06 - 2.06i)T - 19iT^{2} \)
23 \( 1 + 5.51T + 23T^{2} \)
29 \( 1 + 4.89iT - 29T^{2} \)
31 \( 1 + (-3.42 + 3.42i)T - 31iT^{2} \)
37 \( 1 + (7.52 + 7.52i)T + 37iT^{2} \)
41 \( 1 + (3.86 + 3.86i)T + 41iT^{2} \)
43 \( 1 + 1.96iT - 43T^{2} \)
47 \( 1 + (2.05 - 2.05i)T - 47iT^{2} \)
53 \( 1 + 11.4iT - 53T^{2} \)
59 \( 1 + (1.44 - 1.44i)T - 59iT^{2} \)
61 \( 1 + 2.82T + 61T^{2} \)
67 \( 1 + (-8.98 + 8.98i)T - 67iT^{2} \)
71 \( 1 + (8.80 + 8.80i)T + 71iT^{2} \)
73 \( 1 + (-8.70 - 8.70i)T + 73iT^{2} \)
79 \( 1 - 8.87T + 79T^{2} \)
83 \( 1 + (2.45 + 2.45i)T + 83iT^{2} \)
89 \( 1 + (-5.38 + 5.38i)T - 89iT^{2} \)
97 \( 1 + (-2.17 + 2.17i)T - 97iT^{2} \)
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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

−8.928856291419638798801807635574, −8.194153646336959644785826481598, −7.75649373881224662482847955864, −6.63136796635676663420076449806, −5.98839130865403623242303822803, −5.04438315183126156281721132460, −4.05992487785609004371055654642, −3.61279108793213914085276547632, −2.09601049686161425837239503601, −0.34492926941067450622393791157, 1.53330570761089241761289320841, 2.83291557628842597875143343634, 3.54323815899997512720463531480, 4.40246985165215642202273184856, 5.41351449288544394825181205733, 6.44061413840324875197948328227, 6.89617069405304183705604560145, 8.107081790583664904280239143388, 8.764026786467882115351111295310, 9.709952116194645506386898167464

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