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Properties

Label 2-1638-91.74-c1-0-17
Degree $2$
Conductor $1638$
Sign $0.107 - 0.994i$
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 + (2.07 + 3.58i)5-s + (0.321 + 2.62i)7-s + 8-s + (2.07 + 3.58i)10-s + (0.261 + 0.453i)11-s + (−3.28 − 1.47i)13-s + (0.321 + 2.62i)14-s + 16-s + 2.52·17-s + (2.84 − 4.93i)19-s + (2.07 + 3.58i)20-s + (0.261 + 0.453i)22-s + 7.39·23-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.926 + 1.60i)5-s + (0.121 + 0.992i)7-s + 0.353·8-s + (0.654 + 1.13i)10-s + (0.0788 + 0.136i)11-s + (−0.912 − 0.409i)13-s + (0.0859 + 0.701i)14-s + 0.250·16-s + 0.611·17-s + (0.653 − 1.13i)19-s + (0.463 + 0.802i)20-s + (0.0557 + 0.0965i)22-s + 1.54·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.107 - 0.994i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.107 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.186338532\)
\(L(\frac12)\) \(\approx\) \(3.186338532\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.321 - 2.62i)T \)
13 \( 1 + (3.28 + 1.47i)T \)
good5 \( 1 + (-2.07 - 3.58i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.261 - 0.453i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 2.52T + 17T^{2} \)
19 \( 1 + (-2.84 + 4.93i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.39T + 23T^{2} \)
29 \( 1 + (-1.54 + 2.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.17 - 3.76i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.66T + 37T^{2} \)
41 \( 1 + (2.33 - 4.03i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.81 + 8.34i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.58 - 9.66i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.00192 + 0.00332i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.10T + 59T^{2} \)
61 \( 1 + (3.00 - 5.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.61 + 2.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.98 + 6.90i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.99 + 10.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.15 - 2.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.08T + 83T^{2} \)
89 \( 1 - 10.8T + 89T^{2} \)
97 \( 1 + (1.31 + 2.28i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.637142352374731179634466909886, −8.989934861628734234751972983321, −7.59361349978091976167602843168, −7.02511341144515611712621689020, −6.29725804607332262893088603008, −5.44933114370563284342200622480, −4.88957682186544139071905638081, −3.08836228415907569977457078278, −2.93313818146319979993622001394, −1.85527579418516170328351539597, 1.00745658262185297193752728969, 1.86656406625403484517308516219, 3.34830196982673378140095275509, 4.37574093191232176338915094445, 5.10329710062863233496129808085, 5.58733880984985482331794512866, 6.71257474975082390015788069442, 7.52957115348382857326368399179, 8.400408159321325986797438926456, 9.315882622031483360711847825780

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