↑

Properties

Label 2-1638-39.8-c1-0-18
Degree $2$
Conductor $1638$
Sign $0.749 + 0.662i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (2.34 − 2.34i)5-s + (0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + 3.32i·10-s + (−0.324 − 0.324i)11-s + (−0.665 − 3.54i)13-s + 1.00i·14-s − 1.00·16-s + 1.62·17-s + (5.77 + 5.77i)19-s + (−2.34 − 2.34i)20-s + 0.458·22-s − 0.0634·23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (1.05 − 1.05i)5-s + (0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + 1.05i·10-s + (−0.0977 − 0.0977i)11-s + (−0.184 − 0.982i)13-s + 0.267i·14-s − 0.250·16-s + 0.395·17-s + (1.32 + 1.32i)19-s + (−0.525 − 0.525i)20-s + 0.0977·22-s − 0.0132·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.749 + 0.662i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.680096821\)
\(L(\frac12)\) \(\approx\) \(1.680096821\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.665 + 3.54i)T \)
good5 \( 1 + (-2.34 + 2.34i)T - 5iT^{2} \)
11 \( 1 + (0.324 + 0.324i)T + 11iT^{2} \)
17 \( 1 - 1.62T + 17T^{2} \)
19 \( 1 + (-5.77 - 5.77i)T + 19iT^{2} \)
23 \( 1 + 0.0634T + 23T^{2} \)
29 \( 1 + 4.93iT - 29T^{2} \)
31 \( 1 + (-6.43 - 6.43i)T + 31iT^{2} \)
37 \( 1 + (-1.02 + 1.02i)T - 37iT^{2} \)
41 \( 1 + (-1.70 + 1.70i)T - 41iT^{2} \)
43 \( 1 - 2.64iT - 43T^{2} \)
47 \( 1 + (7.39 + 7.39i)T + 47iT^{2} \)
53 \( 1 + 5.78iT - 53T^{2} \)
59 \( 1 + (3.10 + 3.10i)T + 59iT^{2} \)
61 \( 1 - 6.02T + 61T^{2} \)
67 \( 1 + (5.17 + 5.17i)T + 67iT^{2} \)
71 \( 1 + (-2.44 + 2.44i)T - 71iT^{2} \)
73 \( 1 + (6.66 - 6.66i)T - 73iT^{2} \)
79 \( 1 - 0.341T + 79T^{2} \)
83 \( 1 + (-1.32 + 1.32i)T - 83iT^{2} \)
89 \( 1 + (8.87 + 8.87i)T + 89iT^{2} \)
97 \( 1 + (1.56 + 1.56i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.372242360353001660857094019859, −8.270140804829748079301444747311, −8.037430171938808175985624224418, −6.93925193251956816320069027436, −5.81934041995050887820740508995, −5.45613908068487179905493219688, −4.61674716400763283848669150089, −3.20257444819484410120641763142, −1.75597979474864421239097741688, −0.855850867257945694206761384512, 1.35665459498314690347122301792, 2.48697902951883048245578902899, 3.02856607010555174649994278863, 4.42734769536409114394829618989, 5.43361033294688517816195652610, 6.41948828091753337944885026734, 7.07708011034840931450576247192, 7.87012773669807399833994207946, 9.019681324126778015880782641480, 9.548850698840143387082281009018

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