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Properties

Label 2-1638-273.68-c1-0-17
Degree $2$
Conductor $1638$
Sign $-0.541 - 0.840i$
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  + i·2-s − 4-s + (−1.84 + 3.20i)5-s + (2.64 + 0.0512i)7-s − i·8-s + (−3.20 − 1.84i)10-s + (3.50 + 2.02i)11-s + (3.08 − 1.85i)13-s + (−0.0512 + 2.64i)14-s + 16-s + 6.36·17-s + (−3.07 + 1.77i)19-s + (1.84 − 3.20i)20-s + (−2.02 + 3.50i)22-s + 4.48i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.826 + 1.43i)5-s + (0.999 + 0.0193i)7-s − 0.353i·8-s + (−1.01 − 0.584i)10-s + (1.05 + 0.609i)11-s + (0.856 − 0.515i)13-s + (−0.0136 + 0.706i)14-s + 0.250·16-s + 1.54·17-s + (−0.705 + 0.407i)19-s + (0.413 − 0.715i)20-s + (−0.431 + 0.746i)22-s + 0.934i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.541 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.541 - 0.840i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.541 - 0.840i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.797540897\)
\(L(\frac12)\) \(\approx\) \(1.797540897\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (-2.64 - 0.0512i)T \)
13 \( 1 + (-3.08 + 1.85i)T \)
good5 \( 1 + (1.84 - 3.20i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3.50 - 2.02i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 6.36T + 17T^{2} \)
19 \( 1 + (3.07 - 1.77i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.48iT - 23T^{2} \)
29 \( 1 + (-5.32 + 3.07i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.99 + 4.03i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.25T + 37T^{2} \)
41 \( 1 + (0.744 + 1.29i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.09 - 1.90i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.69 - 2.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.29 - 3.63i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 3.93T + 59T^{2} \)
61 \( 1 + (13.3 - 7.67i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.73 + 13.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (8.16 + 4.71i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (8.73 - 5.04i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.583 + 1.01i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.64T + 83T^{2} \)
89 \( 1 + 5.37T + 89T^{2} \)
97 \( 1 + (8.55 + 4.94i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.720659937790190697439693549516, −8.513833305247468814665734441197, −7.82081563191024208903100584276, −7.46800640169873809504836732771, −6.40722421556858368073921593762, −5.90989653102782079978681577742, −4.56566091546603455372981384471, −3.86689371006571783563297124976, −2.94849451646744392688618545460, −1.31976149351379315695559625134, 0.907858825832677892306241926594, 1.46489903515667801425576624362, 3.17552008875299058811941176344, 4.23027668765966636654005230088, 4.60004629462335293836491861063, 5.58292210104806429506788683176, 6.68351135061937756373595237037, 8.001800981963598544749890220965, 8.526809974207843398649854168966, 8.810852709484741702504459469604

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