Properties

Label 2-1638-13.4-c1-0-10
Degree $2$
Conductor $1638$
Sign $-0.702 - 0.711i$
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 + 3.46i·5-s + (0.866 − 0.5i)7-s + 0.999i·8-s + (−1.73 + 2.99i)10-s + (3.46 + 2i)11-s + (−2.59 + 2.5i)13-s + 0.999·14-s + (−0.5 + 0.866i)16-s + (0.232 + 0.401i)17-s + (−0.464 + 0.267i)19-s + (−2.99 + 1.73i)20-s + (1.99 + 3.46i)22-s + (−1.13 + 1.96i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + 1.54i·5-s + (0.327 − 0.188i)7-s + 0.353i·8-s + (−0.547 + 0.948i)10-s + (1.04 + 0.603i)11-s + (−0.720 + 0.693i)13-s + 0.267·14-s + (−0.125 + 0.216i)16-s + (0.0562 + 0.0974i)17-s + (−0.106 + 0.0614i)19-s + (−0.670 + 0.387i)20-s + (0.426 + 0.738i)22-s + (−0.236 + 0.409i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.702 - 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.702 - 0.711i$
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: \((2,\ 1638,\ (\ :1/2),\ -0.702 - 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.398916938\)
\(L(\frac12)\) \(\approx\) \(2.398916938\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 - 2.5i)T \)
good5 \( 1 - 3.46iT - 5T^{2} \)
11 \( 1 + (-3.46 - 2i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.232 - 0.401i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.464 - 0.267i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.13 - 1.96i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4 + 6.92i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.464iT - 31T^{2} \)
37 \( 1 + (7.73 + 4.46i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6 - 3.46i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.23 + 2.13i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.46iT - 47T^{2} \)
53 \( 1 - 11.7T + 53T^{2} \)
59 \( 1 + (8.42 - 4.86i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.59 + 4.5i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.76 - 1.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (10.3 - 5.96i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 2.92iT - 73T^{2} \)
79 \( 1 + 2.53T + 79T^{2} \)
83 \( 1 + 1.73iT - 83T^{2} \)
89 \( 1 + (7.16 + 4.13i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.26 + 4.19i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.807535935344387396369437901338, −8.870534505732894952614931579635, −7.65938833161333710079055272769, −7.18608992486505705371365049580, −6.52721481268586565573078432019, −5.81701853173329280080736364529, −4.51133628571109532771582843935, −3.92076257294677832733492709618, −2.83009841478646071295140190427, −1.90353373327086954508893271009, 0.75656266121290269311934935199, 1.75941581814683348079673352987, 3.12162205656384707986141094911, 4.16962241244482876833634540900, 4.95167586985191917313379168070, 5.49116764581241751495005278717, 6.46716936063112618777092756207, 7.52015229786117856521398375895, 8.677567296440278019246218820633, 8.820575980460263507306200426881

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