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Properties

Label 2-1456-1456.83-c0-0-1
Degree $2$
Conductor $1456$
Sign $0.773 - 0.633i$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
Motivic weight $0$
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 + (0.707 + 0.707i)3-s + 4-s + (−0.707 − 0.707i)6-s + (0.707 − 0.707i)7-s − 8-s + i·11-s + (0.707 + 0.707i)12-s + (0.707 + 0.707i)13-s + (−0.707 + 0.707i)14-s + 16-s − 1.41·19-s + 1.00·21-s − i·22-s + i·23-s + (−0.707 − 0.707i)24-s + ⋯
L(s)  = 1  − 2-s + (0.707 + 0.707i)3-s + 4-s + (−0.707 − 0.707i)6-s + (0.707 − 0.707i)7-s − 8-s + i·11-s + (0.707 + 0.707i)12-s + (0.707 + 0.707i)13-s + (−0.707 + 0.707i)14-s + 16-s − 1.41·19-s + 1.00·21-s − i·22-s + i·23-s + (−0.707 − 0.707i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 - 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $0.773 - 0.633i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1456} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :0),\ 0.773 - 0.633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9867712701\)
\(L(\frac12)\) \(\approx\) \(0.9867712701\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
7 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
5 \( 1 + T^{2} \)
11 \( 1 - iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + 1.41T + T^{2} \)
23 \( 1 - iT - T^{2} \)
29 \( 1 + (-1 - i)T + iT^{2} \)
31 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
43 \( 1 + (-1 + i)T - iT^{2} \)
47 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
67 \( 1 - iT - T^{2} \)
71 \( 1 + (1 - i)T - iT^{2} \)
73 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
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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

−9.838877557085212053648694154284, −8.917662544824756288696583267148, −8.464760208963623475046482217312, −7.55503403997005711339155600577, −6.86009558818959019194903966744, −5.88293789406643173909326527371, −4.38930962708456631559334001453, −3.90202633889522459297871623799, −2.53514851842422573374821974219, −1.48599651148151914626034063595, 1.21707956096429052053591312456, 2.37589077079459094863222053953, 2.97866949300618914643159554121, 4.55354441046586782234358822383, 6.07805567782235101441353558062, 6.26797143514017083991776186054, 7.72530373004662862298966770249, 8.180391173526677818702520865771, 8.504477349188029415160519403261, 9.331217999357274548849672280699

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