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Properties

Label 2-4600-5.4-c1-0-44
Degree $2$
Conductor $4600$
Sign $-0.447 - 0.894i$
Analytic cond. $36.7311$
Root an. cond. $6.06062$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.07i·3-s + 2.07i·7-s − 6.48·9-s + 5.07·11-s − 3.48i·13-s − 6.48i·17-s + 7.48·19-s − 6.40·21-s + i·23-s − 10.7i·27-s + 1.56·29-s + 0.0791·31-s + 15.6i·33-s + 9.71i·37-s + 10.7·39-s + ⋯
L(s)  = 1  + 1.77i·3-s + 0.785i·7-s − 2.16·9-s + 1.53·11-s − 0.965i·13-s − 1.57i·17-s + 1.71·19-s − 1.39·21-s + 0.208i·23-s − 2.06i·27-s + 0.289·29-s + 0.0142·31-s + 2.72i·33-s + 1.59i·37-s + 1.71·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(36.7311\)
Root analytic conductor: \(6.06062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4600} (4049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4600,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.176958687\)
\(L(\frac12)\) \(\approx\) \(2.176958687\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 - iT \)
good3 \( 1 - 3.07iT - 3T^{2} \)
7 \( 1 - 2.07iT - 7T^{2} \)
11 \( 1 - 5.07T + 11T^{2} \)
13 \( 1 + 3.48iT - 13T^{2} \)
17 \( 1 + 6.48iT - 17T^{2} \)
19 \( 1 - 7.48T + 19T^{2} \)
29 \( 1 - 1.56T + 29T^{2} \)
31 \( 1 - 0.0791T + 31T^{2} \)
37 \( 1 - 9.71iT - 37T^{2} \)
41 \( 1 + 0.480T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + 6.96iT - 47T^{2} \)
53 \( 1 - 11.7iT - 53T^{2} \)
59 \( 1 - 11.5T + 59T^{2} \)
61 \( 1 - 7.88T + 61T^{2} \)
67 \( 1 - 9.71iT - 67T^{2} \)
71 \( 1 + 9.67T + 71T^{2} \)
73 \( 1 + 13.2iT - 73T^{2} \)
79 \( 1 - 12.3T + 79T^{2} \)
83 \( 1 + 4.59iT - 83T^{2} \)
89 \( 1 + 8.31T + 89T^{2} \)
97 \( 1 + 7.23iT - 97T^{2} \)
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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

−8.935996983105362456161937020306, −8.047941271962651953959156250626, −7.09257529941082758849995709858, −6.10917256979237677979547879655, −5.34681743300551116413436745076, −4.95037052750560516897094422676, −4.05187724026678254465706697962, −3.20765812373988208530851551231, −2.76742338734489608704131622068, −0.996754080179682520150237908565, 0.78827300348189271728511041563, 1.47575122214460229612176742393, 2.21204871927215847940364126987, 3.56210284994336987360116243772, 4.07876165590233436536087369665, 5.43841758223119210162516531326, 6.15913775270643562278263504133, 6.91630488626717906698764013698, 7.07562599537961612120530400612, 7.953417865591742137987763846865

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