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Properties

Label 2-1638-13.4-c1-0-14
Degree $2$
Conductor $1638$
Sign $0.994 + 0.105i$
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 − 0.901i·5-s + (0.866 − 0.5i)7-s − 0.999i·8-s + (−0.450 + 0.781i)10-s + (3.75 + 2.16i)11-s + (−0.426 + 3.58i)13-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (−2.53 − 4.38i)17-s + (5.34 − 3.08i)19-s + (0.781 − 0.450i)20-s + (−2.16 − 3.75i)22-s + (−4.22 + 7.31i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s − 0.403i·5-s + (0.327 − 0.188i)7-s − 0.353i·8-s + (−0.142 + 0.246i)10-s + (1.13 + 0.653i)11-s + (−0.118 + 0.992i)13-s − 0.267·14-s + (−0.125 + 0.216i)16-s + (−0.614 − 1.06i)17-s + (1.22 − 0.708i)19-s + (0.174 − 0.100i)20-s + (−0.462 − 0.800i)22-s + (−0.881 + 1.52i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.994 + 0.105i$
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: \((2,\ 1638,\ (\ :1/2),\ 0.994 + 0.105i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.387551326\)
\(L(\frac12)\) \(\approx\) \(1.387551326\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.426 - 3.58i)T \)
good5 \( 1 + 0.901iT - 5T^{2} \)
11 \( 1 + (-3.75 - 2.16i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.53 + 4.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.34 + 3.08i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.22 - 7.31i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.09 - 1.89i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.873iT - 31T^{2} \)
37 \( 1 + (-0.124 - 0.0721i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.46 - 1.99i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.85 - 6.67i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.92iT - 47T^{2} \)
53 \( 1 + 1.69T + 53T^{2} \)
59 \( 1 + (-7.40 + 4.27i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.16 + 7.21i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.99 + 5.19i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.83 + 1.63i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 0.539iT - 73T^{2} \)
79 \( 1 - 6.53T + 79T^{2} \)
83 \( 1 - 13.2iT - 83T^{2} \)
89 \( 1 + (-6.74 - 3.89i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.1 + 5.85i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.364754563557655780406274091985, −8.924328192734130654379720938683, −7.70932413740074769062901828798, −7.17478549824857992082557148038, −6.38265313561375339385787752714, −5.03735024868166052596681010909, −4.36811206288394672459834353553, −3.29634391080080342716618316358, −1.99903119263241178157146622856, −1.05992034725063246260820329733, 0.842534623739461893395076219275, 2.16079708030022018615363977058, 3.36485186679785775082858627651, 4.37120304835677143173640720982, 5.67877965827264367883147574496, 6.14701539415968648002954986691, 7.05829557173326216903647603756, 7.916731067416108357431290060827, 8.591441250887337449988616805023, 9.193650114727170519840168576555

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