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Properties

Label 2-1638-13.4-c1-0-7
Degree $2$
Conductor $1638$
Sign $0.923 - 0.383i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s − 4.11i·5-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + (−2.05 + 3.56i)10-s + (5.41 + 3.12i)11-s + (1.33 + 3.34i)13-s + 0.999·14-s + (−0.5 + 0.866i)16-s + (2.23 + 3.86i)17-s + (−2.94 + 1.70i)19-s + (3.56 − 2.05i)20-s + (−3.12 − 5.41i)22-s + (−1.49 + 2.58i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s − 1.84i·5-s + (−0.327 + 0.188i)7-s − 0.353i·8-s + (−0.651 + 1.12i)10-s + (1.63 + 0.942i)11-s + (0.371 + 0.928i)13-s + 0.267·14-s + (−0.125 + 0.216i)16-s + (0.541 + 0.938i)17-s + (−0.675 + 0.390i)19-s + (0.797 − 0.460i)20-s + (−0.666 − 1.15i)22-s + (−0.311 + 0.539i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.923 - 0.383i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.923 - 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.110267906\)
\(L(\frac12)\) \(\approx\) \(1.110267906\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-1.33 - 3.34i)T \)
good5 \( 1 + 4.11iT - 5T^{2} \)
11 \( 1 + (-5.41 - 3.12i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.23 - 3.86i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.94 - 1.70i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.49 - 2.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.51 - 4.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.7iT - 31T^{2} \)
37 \( 1 + (-3.52 - 2.03i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.96 + 3.44i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.83 + 3.17i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 9.29iT - 47T^{2} \)
53 \( 1 - 1.32T + 53T^{2} \)
59 \( 1 + (-2.81 + 1.62i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.550 + 0.953i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.360 - 0.208i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-13.1 + 7.61i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.7iT - 73T^{2} \)
79 \( 1 - 1.31T + 79T^{2} \)
83 \( 1 - 8.04iT - 83T^{2} \)
89 \( 1 + (8.60 + 4.96i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.5 - 6.07i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.432977419635513497496273459163, −8.673360464868108279538611960780, −8.319607208153126650979382619925, −7.06760343098430974736637467441, −6.31278656965280178936248882938, −5.24399175902788642328552978285, −4.22903852715136845753321174626, −3.70141404359133002302889539655, −1.68243821042084537098837174117, −1.39958233187838781177203090678, 0.56128592125252866826662667249, 2.30870770245819443981073822885, 3.26399926745146487215483505997, 4.01269597838947612820567713090, 5.75353791518343012983356947254, 6.31248416180033278478115554253, 6.85377869684081747328917692489, 7.67833455598890971352795063347, 8.432761849569828953929751523503, 9.532709239073225586928136273249

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