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Properties

Label 2-1638-13.4-c1-0-1
Degree $2$
Conductor $1638$
Sign $0.0515 - 0.998i$
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 − 2.73i·5-s + (−0.866 + 0.5i)7-s − 0.999i·8-s + (−1.36 + 2.36i)10-s + (−1.5 − 0.866i)11-s + (−3.59 − 0.232i)13-s + 0.999·14-s + (−0.5 + 0.866i)16-s + (−0.133 − 0.232i)17-s + (−0.866 + 0.5i)19-s + (2.36 − 1.36i)20-s + (0.866 + 1.5i)22-s + (−1.73 + 3i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s − 1.22i·5-s + (−0.327 + 0.188i)7-s − 0.353i·8-s + (−0.431 + 0.748i)10-s + (−0.452 − 0.261i)11-s + (−0.997 − 0.0643i)13-s + 0.267·14-s + (−0.125 + 0.216i)16-s + (−0.0324 − 0.0562i)17-s + (−0.198 + 0.114i)19-s + (0.529 − 0.305i)20-s + (0.184 + 0.319i)22-s + (−0.361 + 0.625i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0515 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0515 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.0515 - 0.998i$
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: \((2,\ 1638,\ (\ :1/2),\ 0.0515 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3037912322\)
\(L(\frac12)\) \(\approx\) \(0.3037912322\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.59 + 0.232i)T \)
good5 \( 1 + 2.73iT - 5T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.133 + 0.232i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.73 - 3i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.232 - 0.401i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8.19iT - 31T^{2} \)
37 \( 1 + (2.83 + 1.63i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.59 - 1.5i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.36 - 5.83i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 4.46iT - 47T^{2} \)
53 \( 1 + 7T + 53T^{2} \)
59 \( 1 + (11.1 - 6.46i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.59 - 4.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.26 - 2.46i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.09 + 4.09i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 1.46iT - 73T^{2} \)
79 \( 1 - 15.9T + 79T^{2} \)
83 \( 1 + 10.1iT - 83T^{2} \)
89 \( 1 + (3.06 + 1.76i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.43 + 0.830i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.407970166554432703057398956086, −8.948583028908898092779748173376, −8.111190889429782803406423074805, −7.47281806995968213634845871003, −6.40829989080868768616117329519, −5.34070015264298279430265884688, −4.69257207666598838490961802242, −3.49525555904957893082338120066, −2.41310867188804024984334067339, −1.20931236325008293056005258602, 0.14828689520451778951719055200, 2.13941053305896401936935349682, 2.87819728676688643803261904369, 4.11825568629710228907554964164, 5.23590409029145978157722946742, 6.28088336786786603251600828170, 6.84061922299957008908848472846, 7.57776402313324679074127013781, 8.184006057328682855685433038098, 9.456446121383409796323187591175

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