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Properties

Label 2-1638-13.10-c1-0-21
Degree $2$
Conductor $1638$
Sign $0.545 + 0.838i$
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 + 1.56i·5-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (0.781 + 1.35i)10-s + (2.48 − 1.43i)11-s + (2.99 − 2.00i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (1.11 − 1.93i)17-s + (−6.26 − 3.61i)19-s + (1.35 + 0.781i)20-s + (1.43 − 2.48i)22-s + (0.833 + 1.44i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.699i·5-s + (−0.327 − 0.188i)7-s − 0.353i·8-s + (0.247 + 0.428i)10-s + (0.748 − 0.432i)11-s + (0.830 − 0.556i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.270 − 0.468i)17-s + (−1.43 − 0.830i)19-s + (0.302 + 0.174i)20-s + (0.305 − 0.529i)22-s + (0.173 + 0.301i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.545 + 0.838i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.545 + 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.483004035\)
\(L(\frac12)\) \(\approx\) \(2.483004035\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.99 + 2.00i)T \)
good5 \( 1 - 1.56iT - 5T^{2} \)
11 \( 1 + (-2.48 + 1.43i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.11 + 1.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.26 + 3.61i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.833 - 1.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.41 - 4.18i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.597iT - 31T^{2} \)
37 \( 1 + (-0.0333 + 0.0192i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.88 + 3.97i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.04 + 8.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.02iT - 47T^{2} \)
53 \( 1 - 5.98T + 53T^{2} \)
59 \( 1 + (0.776 + 0.448i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.12 + 12.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.42 - 0.820i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.98 - 1.14i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 11.2iT - 73T^{2} \)
79 \( 1 - 4.26T + 79T^{2} \)
83 \( 1 + 4.94iT - 83T^{2} \)
89 \( 1 + (2.09 - 1.21i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.23 + 2.44i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.229709581906116679830668668932, −8.649430865346620060428415471582, −7.43497358279463259086057394714, −6.64082985295823651550907442276, −6.11770329685873681563835628279, −5.08985064098628792087465970329, −4.00563555074576647604197488362, −3.30308087923947761213260794989, −2.39991897578885041132813792532, −0.884122202246548482612109342596, 1.32438888429897324348160615643, 2.57009290462463113762655907093, 4.05969386808096150298022597000, 4.23085033681988316625214423131, 5.51042994829537604547785030778, 6.27608401239095868569491536960, 6.81394036175540004407261590616, 8.037356086098102184663701297798, 8.607544936891610342749944323932, 9.325682861885630036064654778026

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