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Properties

Label 2-2775-555.554-c0-0-4
Degree $2$
Conductor $2775$
Sign $-0.447 + 0.894i$
Analytic cond. $1.38490$
Root an. cond. $1.17682$
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  − i·3-s + 4-s − 2i·7-s − 9-s − i·12-s + 16-s − 2·21-s + i·27-s − 2i·28-s − 36-s + i·37-s − i·48-s − 3·49-s + 2i·63-s + 64-s + ⋯
L(s)  = 1  − i·3-s + 4-s − 2i·7-s − 9-s − i·12-s + 16-s − 2·21-s + i·27-s − 2i·28-s − 36-s + i·37-s − i·48-s − 3·49-s + 2i·63-s + 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2775\)    =    \(3 \cdot 5^{2} \cdot 37\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(1.38490\)
Root analytic conductor: \(1.17682\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2775} (2774, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2775,\ (\ :0),\ -0.447 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.470457906\)
\(L(\frac12)\) \(\approx\) \(1.470457906\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
37 \( 1 - iT \)
good2 \( 1 - T^{2} \)
7 \( 1 + 2iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 2iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
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

−8.359436938150970297570699482944, −7.78169779447524737277763241323, −7.15985191432586972939484790405, −6.71445813245961913022405014925, −6.01407364121643980287606356339, −4.87907079122531678846271638823, −3.76385792528669482693733357439, −2.97620720409812134232405578495, −1.79715951110790924092822455953, −0.927613096332938647775603914877, 1.98548331413407362276180711636, 2.70226732970003992426595199101, 3.43357047848899628505736781922, 4.65772101760165123298306839935, 5.64374630926559220231968422013, 5.84238916896418778329887318492, 6.81553199700463675489344403482, 7.926774885568900980806729926140, 8.602331946771432747992462363482, 9.236109315718599368052337129613

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