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Properties

Label 2-1472-1472.413-c0-0-1
Degree $2$
Conductor $1472$
Sign $0.881 + 0.471i$
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  + (−0.382 + 0.923i)2-s + (−0.382 − 0.0761i)3-s + (−0.707 − 0.707i)4-s + (0.216 − 0.324i)6-s + (0.923 − 0.382i)8-s + (−0.783 − 0.324i)9-s + (0.216 + 0.324i)12-s + (−0.216 − 0.324i)13-s + i·16-s + (0.599 − 0.599i)18-s + (0.382 − 0.923i)23-s + (−0.382 + 0.0761i)24-s + (−0.382 − 0.923i)25-s + (0.382 − 0.0761i)26-s + (0.599 + 0.400i)27-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)2-s + (−0.382 − 0.0761i)3-s + (−0.707 − 0.707i)4-s + (0.216 − 0.324i)6-s + (0.923 − 0.382i)8-s + (−0.783 − 0.324i)9-s + (0.216 + 0.324i)12-s + (−0.216 − 0.324i)13-s + i·16-s + (0.599 − 0.599i)18-s + (0.382 − 0.923i)23-s + (−0.382 + 0.0761i)24-s + (−0.382 − 0.923i)25-s + (0.382 − 0.0761i)26-s + (0.599 + 0.400i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1472\)    =    \(2^{6} \cdot 23\)
Sign: $0.881 + 0.471i$
Analytic conductor: \(0.734623\)
Root analytic conductor: \(0.857101\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1472} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1472,\ (\ :0),\ 0.881 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5649418583\)
\(L(\frac12)\) \(\approx\) \(0.5649418583\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 - 0.923i)T \)
23 \( 1 + (-0.382 + 0.923i)T \)
good3 \( 1 + (0.382 + 0.0761i)T + (0.923 + 0.382i)T^{2} \)
5 \( 1 + (0.382 + 0.923i)T^{2} \)
7 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (0.923 - 0.382i)T^{2} \)
13 \( 1 + (0.216 + 0.324i)T + (-0.382 + 0.923i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-0.382 + 0.923i)T^{2} \)
29 \( 1 + (-0.324 + 1.63i)T + (-0.923 - 0.382i)T^{2} \)
31 \( 1 - 0.765iT - T^{2} \)
37 \( 1 + (-0.382 - 0.923i)T^{2} \)
41 \( 1 + (-0.765 + 1.84i)T + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + (-0.923 + 0.382i)T^{2} \)
47 \( 1 + (1 + i)T + iT^{2} \)
53 \( 1 + (0.923 - 0.382i)T^{2} \)
59 \( 1 + (1.08 - 1.63i)T + (-0.382 - 0.923i)T^{2} \)
61 \( 1 + (-0.923 - 0.382i)T^{2} \)
67 \( 1 + (-0.923 - 0.382i)T^{2} \)
71 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 + (-0.382 + 0.923i)T^{2} \)
89 \( 1 + (0.707 - 0.707i)T^{2} \)
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

−9.502773636445319094921367300633, −8.636184898156146022643568278194, −8.133821140127717741130988127969, −7.12420939440511903529401135306, −6.38031355919030160921873671676, −5.70769083875309325573594794541, −4.89940591060155826179476359165, −3.87651153937540560657556111275, −2.41518409875495160665413968061, −0.58323883456524963885411174514, 1.41711752381582801047826787520, 2.68797926960873314046303226804, 3.54444734333588368451289385143, 4.72488352614317995457831663074, 5.38625195644639512617372997267, 6.52527687308122578185593793519, 7.64264638962786805602653356793, 8.226173239665050781684233710546, 9.324167336161224357393237010030, 9.582006840586143901858827412337

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