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Properties

Label 2-1472-368.45-c0-0-2
Degree $2$
Conductor $1472$
Sign $-0.923 + 0.382i$
Analytic cond. $0.734623$
Root an. cond. $0.857101$
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 − i)3-s + i·9-s + (−1 − i)13-s − i·23-s − i·25-s + (−1 − i)29-s + 2i·39-s + 2i·41-s − 2·47-s − 49-s + (1 − i)59-s + (−1 + i)69-s − 2i·73-s + (−1 + i)75-s + 81-s + ⋯
L(s)  = 1  + (−1 − i)3-s + i·9-s + (−1 − i)13-s − i·23-s − i·25-s + (−1 − i)29-s + 2i·39-s + 2i·41-s − 2·47-s − 49-s + (1 − i)59-s + (−1 + i)69-s − 2i·73-s + (−1 + i)75-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1472\)    =    \(2^{6} \cdot 23\)
Sign: $-0.923 + 0.382i$
Analytic conductor: \(0.734623\)
Root analytic conductor: \(0.857101\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1472} (1425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1472,\ (\ :0),\ -0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4781549006\)
\(L(\frac12)\) \(\approx\) \(0.4781549006\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.540864728123536893037153029622, −8.146934818605474248307196468854, −7.79131131330721310153342056583, −6.71659709475276646863850361219, −6.24395529802743671147969119772, −5.31232219173448931934722221886, −4.54731029416957891172573435628, −3.05080532227281323577501297150, −1.90870223265964331687850674993, −0.42333670295963037680140036137, 1.84990701873601891355084067052, 3.41319010040791130690878703668, 4.27555675899622033603690998351, 5.17504049756295581018714727881, 5.61806620553868395572879005222, 6.82033105735933499790066607771, 7.41861745556457510078669580158, 8.691518392501832098574446649930, 9.572983584122506562908521870706, 9.891766602840267351340748700194

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