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Properties

Label 2-1638-91.81-c1-0-29
Degree $2$
Conductor $1638$
Sign $0.649 + 0.760i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.10 − 1.91i)5-s + (1.44 − 2.21i)7-s + 0.999·8-s + 2.20·10-s + 1.05·11-s + (3.18 + 1.69i)13-s + (1.19 + 2.36i)14-s + (−0.5 + 0.866i)16-s + (−0.472 − 0.817i)17-s + 3.92·19-s + (−1.10 + 1.91i)20-s + (−0.527 + 0.914i)22-s + (−3.11 + 5.39i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.493 − 0.854i)5-s + (0.547 − 0.836i)7-s + 0.353·8-s + 0.697·10-s + 0.318·11-s + (0.883 + 0.469i)13-s + (0.318 + 0.631i)14-s + (−0.125 + 0.216i)16-s + (−0.114 − 0.198i)17-s + 0.901·19-s + (−0.246 + 0.427i)20-s + (−0.112 + 0.194i)22-s + (−0.649 + 1.12i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.649 + 0.760i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.649 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.320925813\)
\(L(\frac12)\) \(\approx\) \(1.320925813\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-1.44 + 2.21i)T \)
13 \( 1 + (-3.18 - 1.69i)T \)
good5 \( 1 + (1.10 + 1.91i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 1.05T + 11T^{2} \)
17 \( 1 + (0.472 + 0.817i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 3.92T + 19T^{2} \)
23 \( 1 + (3.11 - 5.39i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.888 + 1.53i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.63 + 6.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.13 - 1.95i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.63 - 2.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.537 + 0.930i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.42 + 4.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.94 + 8.55i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.509 + 0.882i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 0.0764T + 61T^{2} \)
67 \( 1 + 11.7T + 67T^{2} \)
71 \( 1 + (-4.20 + 7.28i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.57 + 11.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.00 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.77T + 83T^{2} \)
89 \( 1 + (-6.66 + 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.99 - 15.5i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.203212751124845635153453044008, −8.255663855367147639824515804347, −7.86930024233833087197680150840, −7.01153783482937677081041817849, −6.10821894626824446900804552739, −5.13717082420092944438437834970, −4.34014424599343051823591332171, −3.61907307423262724107495622974, −1.67848482206403403316671416650, −0.67681165440161443014063186229, 1.25632902704309532296930812041, 2.54793082671681643617138536730, 3.31009827053380924218013627012, 4.26516968696598856618283651027, 5.38592839600448365493122230469, 6.31937781908099984363615296515, 7.24066595646403219068177578610, 8.107070310467273625243279210379, 8.676149782668870829545189089332, 9.458670336004946972335572305038

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