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Properties

Label 2-1632-408.275-c0-0-0
Degree $2$
Conductor $1632$
Sign $0.982 - 0.185i$
Analytic cond. $0.814474$
Root an. cond. $0.902482$
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  + (−0.923 + 0.382i)3-s + (0.707 − 0.707i)9-s + (0.382 − 0.0761i)11-s + (−0.923 + 0.382i)17-s + (1.30 − 0.541i)19-s + (0.382 − 0.923i)25-s + (−0.382 + 0.923i)27-s + (−0.324 + 0.216i)33-s + (1.63 + 1.08i)41-s + (1.70 + 0.707i)43-s + (−0.382 − 0.923i)49-s + (0.707 − 0.707i)51-s + (−0.999 + i)57-s + (0.707 + 0.292i)59-s − 0.765i·67-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)3-s + (0.707 − 0.707i)9-s + (0.382 − 0.0761i)11-s + (−0.923 + 0.382i)17-s + (1.30 − 0.541i)19-s + (0.382 − 0.923i)25-s + (−0.382 + 0.923i)27-s + (−0.324 + 0.216i)33-s + (1.63 + 1.08i)41-s + (1.70 + 0.707i)43-s + (−0.382 − 0.923i)49-s + (0.707 − 0.707i)51-s + (−0.999 + i)57-s + (0.707 + 0.292i)59-s − 0.765i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1632 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1632 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1632\)    =    \(2^{5} \cdot 3 \cdot 17\)
Sign: $0.982 - 0.185i$
Analytic conductor: \(0.814474\)
Root analytic conductor: \(0.902482\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1632} (1295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1632,\ (\ :0),\ 0.982 - 0.185i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8624164074\)
\(L(\frac12)\) \(\approx\) \(0.8624164074\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.923 - 0.382i)T \)
17 \( 1 + (0.923 - 0.382i)T \)
good5 \( 1 + (-0.382 + 0.923i)T^{2} \)
7 \( 1 + (0.382 + 0.923i)T^{2} \)
11 \( 1 + (-0.382 + 0.0761i)T + (0.923 - 0.382i)T^{2} \)
13 \( 1 - iT^{2} \)
19 \( 1 + (-1.30 + 0.541i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (0.923 - 0.382i)T^{2} \)
29 \( 1 + (0.382 - 0.923i)T^{2} \)
31 \( 1 + (0.923 + 0.382i)T^{2} \)
37 \( 1 + (-0.923 - 0.382i)T^{2} \)
41 \( 1 + (-1.63 - 1.08i)T + (0.382 + 0.923i)T^{2} \)
43 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (-0.382 - 0.923i)T^{2} \)
67 \( 1 + 0.765iT - T^{2} \)
71 \( 1 + (0.923 + 0.382i)T^{2} \)
73 \( 1 + (-1.08 - 1.63i)T + (-0.382 + 0.923i)T^{2} \)
79 \( 1 + (0.923 - 0.382i)T^{2} \)
83 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
89 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
97 \( 1 + (-0.923 + 0.617i)T + (0.382 - 0.923i)T^{2} \)
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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

−9.600336974735635029684594566463, −9.050414965951117519385872598497, −7.974525699056288860264985693085, −7.00164269843890063952466621580, −6.34505515343691308582823865620, −5.53458502890493361681620197451, −4.62809361040331841330862358756, −3.92887703275107201337766963304, −2.64916212797053662123735894505, −1.03927451971676922864204095994, 1.08365086257510234327917531632, 2.35838594150533130781150142647, 3.74503445541105524922902719110, 4.72527521259776247305043024610, 5.55914979406971260641946321971, 6.25873188816764666341312888772, 7.25746059267950051257419550098, 7.60071236589455921255113300593, 8.911493992503629546696219860096, 9.520614508851589518183132790059

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