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Properties

Label 2-1664-8.5-c1-0-36
Degree $2$
Conductor $1664$
Sign $0.707 + 0.707i$
Analytic cond. $13.2871$
Root an. cond. $3.64514$
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·3-s − i·5-s + 7-s + 2·9-s − 2i·11-s − i·13-s + 15-s + 17-s − 8i·19-s + i·21-s − 6·23-s + 4·25-s + 5i·27-s − 6i·29-s − 8·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.447i·5-s + 0.377·7-s + 0.666·9-s − 0.603i·11-s − 0.277i·13-s + 0.258·15-s + 0.242·17-s − 1.83i·19-s + 0.218i·21-s − 1.25·23-s + 0.800·25-s + 0.962i·27-s − 1.11i·29-s − 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1664\)    =    \(2^{7} \cdot 13\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(13.2871\)
Root analytic conductor: \(3.64514\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1664} (833, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1664,\ (\ :1/2),\ 0.707 + 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.745482547\)
\(L(\frac12)\) \(\approx\) \(1.745482547\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + iT \)
good3 \( 1 - iT - 3T^{2} \)
5 \( 1 + iT - 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
17 \( 1 - T + 17T^{2} \)
19 \( 1 + 8iT - 19T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + iT - 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 - 11T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 + 8iT - 59T^{2} \)
61 \( 1 + 4iT - 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 16T + 89T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−9.331892432206167465861991246655, −8.568671148534305438260605817712, −7.75599201746826426681180659747, −6.92394040243523912543234214978, −5.87825800194947299399718773314, −5.00577761064076616128157476814, −4.35309625564888947535429882951, −3.41645481155897147221232419757, −2.15215473781814847052945632546, −0.71185507079481932750567147194, 1.40767421630135796201242813457, 2.17159586073740107029659073397, 3.58586344505194442386598457845, 4.36667835511829713219442516295, 5.51935645595199180280505350367, 6.31731037302782422460517205538, 7.29143114249156445430887185001, 7.59431958140453135355410494866, 8.569574200849123439207372775387, 9.516485590914282840022580249494

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