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Properties

Label 2-1664-104.51-c0-0-3
Degree $2$
Conductor $1664$
Sign $0.707 - 0.707i$
Analytic cond. $0.830444$
Root an. cond. $0.911287$
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  + 1.41·7-s − 9-s + 1.41i·11-s + i·13-s + 1.41i·19-s − 25-s − 2i·29-s + 1.41·31-s + 1.41·47-s + 1.00·49-s − 1.41i·59-s − 1.41·63-s + 1.41i·67-s − 1.41·71-s + 2.00i·77-s + ⋯
L(s)  = 1  + 1.41·7-s − 9-s + 1.41i·11-s + i·13-s + 1.41i·19-s − 25-s − 2i·29-s + 1.41·31-s + 1.41·47-s + 1.00·49-s − 1.41i·59-s − 1.41·63-s + 1.41i·67-s − 1.41·71-s + 2.00i·77-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1664\)    =    \(2^{7} \cdot 13\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1664} (831, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1664,\ (\ :0),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.183876790\)
\(L(\frac12)\) \(\approx\) \(1.183876790\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.729274669091944238621971207675, −8.753976925674363593866144056459, −7.991118778875799680720779718116, −7.53537767987558127012416354045, −6.34945754145033074110725898502, −5.58905413434534649934704198130, −4.56704443734888087853218613187, −4.04997147785274526714370036907, −2.41170408092711347789188503384, −1.69582877733080275884475994879, 0.993342333303775654043931422512, 2.55268209959452218467104169791, 3.33655909384933682946083809137, 4.64771000980903130132680604758, 5.41039724343645874165461932089, 6.00097525703208075934377615667, 7.19705614773243062308290403851, 8.114273102960184603682855489142, 8.527440484766144276764545979155, 9.184655078846660781536692107606

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