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Properties

Label 2-1638-91.83-c1-0-16
Degree $2$
Conductor $1638$
Sign $0.722 + 0.691i$
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  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−1.15 − 1.15i)5-s + (−1.70 + 2.02i)7-s + (0.707 + 0.707i)8-s + 1.63·10-s + (−1.37 − 1.37i)11-s + (1.50 + 3.27i)13-s + (−0.222 − 2.63i)14-s − 1.00·16-s − 1.50·17-s + (0.934 + 0.934i)19-s + (−1.15 + 1.15i)20-s + 1.95·22-s − 4.19i·23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.517 − 0.517i)5-s + (−0.645 + 0.763i)7-s + (0.250 + 0.250i)8-s + 0.517·10-s + (−0.415 − 0.415i)11-s + (0.416 + 0.909i)13-s + (−0.0593 − 0.704i)14-s − 0.250·16-s − 0.365·17-s + (0.214 + 0.214i)19-s + (−0.258 + 0.258i)20-s + 0.415·22-s − 0.874i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.722 + 0.691i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.722 + 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7407932956\)
\(L(\frac12)\) \(\approx\) \(0.7407932956\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
7 \( 1 + (1.70 - 2.02i)T \)
13 \( 1 + (-1.50 - 3.27i)T \)
good5 \( 1 + (1.15 + 1.15i)T + 5iT^{2} \)
11 \( 1 + (1.37 + 1.37i)T + 11iT^{2} \)
17 \( 1 + 1.50T + 17T^{2} \)
19 \( 1 + (-0.934 - 0.934i)T + 19iT^{2} \)
23 \( 1 + 4.19iT - 23T^{2} \)
29 \( 1 - 0.406T + 29T^{2} \)
31 \( 1 + (-2.31 - 2.31i)T + 31iT^{2} \)
37 \( 1 + (-0.257 - 0.257i)T + 37iT^{2} \)
41 \( 1 + (-3.60 - 3.60i)T + 41iT^{2} \)
43 \( 1 + 2.46iT - 43T^{2} \)
47 \( 1 + (1.41 - 1.41i)T - 47iT^{2} \)
53 \( 1 + 13.4T + 53T^{2} \)
59 \( 1 + (-8.75 + 8.75i)T - 59iT^{2} \)
61 \( 1 + 11.7iT - 61T^{2} \)
67 \( 1 + (-11.1 + 11.1i)T - 67iT^{2} \)
71 \( 1 + (-5.18 + 5.18i)T - 71iT^{2} \)
73 \( 1 + (-2.56 + 2.56i)T - 73iT^{2} \)
79 \( 1 - 1.42T + 79T^{2} \)
83 \( 1 + (-4.90 - 4.90i)T + 83iT^{2} \)
89 \( 1 + (-2.03 + 2.03i)T - 89iT^{2} \)
97 \( 1 + (10.2 + 10.2i)T + 97iT^{2} \)
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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

−9.221695665950232093572737970149, −8.346810849961613277890356125984, −8.048742536087392673672586981974, −6.67497456373415626116444740577, −6.33595644311558897869570000566, −5.24370501945295231475269409813, −4.43037296038538881252819472306, −3.25509154993867560627974788831, −2.02318067733184076198186367393, −0.42157615334557521590622431412, 0.980660032658695935067248992698, 2.56111020932941331906763192037, 3.41563873544095110495727126411, 4.12985113340069312294027180583, 5.36597891540436197735218795437, 6.48382245239529513648426695384, 7.35144864183377966686481166448, 7.75846242026024046588237493555, 8.738428888525264954184309713044, 9.707010987009710216456039716252

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