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Properties

Label 2-1638-91.16-c1-0-10
Degree $2$
Conductor $1638$
Sign $-0.540 - 0.841i$
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  + 2-s + 4-s + (−1.74 + 3.02i)5-s + (2.07 + 1.64i)7-s + 8-s + (−1.74 + 3.02i)10-s + (−2.23 + 3.87i)11-s + (3.57 + 0.498i)13-s + (2.07 + 1.64i)14-s + 16-s + 0.785·17-s + (−1.05 − 1.81i)19-s + (−1.74 + 3.02i)20-s + (−2.23 + 3.87i)22-s − 5.42·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.781 + 1.35i)5-s + (0.784 + 0.620i)7-s + 0.353·8-s + (−0.552 + 0.956i)10-s + (−0.674 + 1.16i)11-s + (0.990 + 0.138i)13-s + (0.554 + 0.438i)14-s + 0.250·16-s + 0.190·17-s + (−0.240 − 0.417i)19-s + (−0.390 + 0.676i)20-s + (−0.477 + 0.826i)22-s − 1.13·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.540 - 0.841i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.540 - 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.136920376\)
\(L(\frac12)\) \(\approx\) \(2.136920376\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 + (-2.07 - 1.64i)T \)
13 \( 1 + (-3.57 - 0.498i)T \)
good5 \( 1 + (1.74 - 3.02i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.23 - 3.87i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 0.785T + 17T^{2} \)
19 \( 1 + (1.05 + 1.81i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.42T + 23T^{2} \)
29 \( 1 + (1.89 + 3.28i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.726 + 1.25i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.64T + 37T^{2} \)
41 \( 1 + (2.68 + 4.64i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.81 - 6.60i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.47 - 4.29i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.81 - 8.33i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 7.87T + 59T^{2} \)
61 \( 1 + (-7.52 - 13.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.91 + 3.31i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.14 + 1.97i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.44 + 5.97i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.08 - 8.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 + 5.91T + 89T^{2} \)
97 \( 1 + (-8.56 + 14.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.930193138537823677958516344080, −8.639063434618876055890152833545, −7.80590283726248100530339775099, −7.29054191973737869722661034962, −6.39499295762909275784705313050, −5.59909959419678431183471251351, −4.52903715675121778220420869880, −3.81832114515784329995348522807, −2.74732370088793941282322010764, −1.94880669876024714930383689703, 0.63790719498207106630153512530, 1.78158132572968363001359609832, 3.55715768778011462438986795021, 3.93833070103406763152157269355, 5.07911506531195835714839972238, 5.46187007192286325548776427003, 6.59425906392702322141983087991, 7.75673865141084616022533075408, 8.308305171756042566589951452599, 8.668586682197112227357070953525

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