Properties

Label 2-4600-5.4-c1-0-85
Degree $2$
Conductor $4600$
Sign $-0.894 + 0.447i$
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  − 0.794i·3-s − 2.47i·7-s + 2.36·9-s − 2.29·11-s − 3.84i·13-s + 7.74i·17-s + 2.29·19-s − 1.96·21-s i·23-s − 4.26i·27-s − 5.28·29-s − 6.40·31-s + 1.82i·33-s − 8.56i·37-s − 3.05·39-s + ⋯
L(s)  = 1  − 0.458i·3-s − 0.934i·7-s + 0.789·9-s − 0.693·11-s − 1.06i·13-s + 1.87i·17-s + 0.527·19-s − 0.428·21-s − 0.208i·23-s − 0.821i·27-s − 0.981·29-s − 1.14·31-s + 0.318i·33-s − 1.40i·37-s − 0.488·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(4600\)    =    \(2^{3} \cdot 5^{2} \cdot 23\)
Sign: $-0.894 + 0.447i$
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: \((2,\ 4600,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.224049062\)
\(L(\frac12)\) \(\approx\) \(1.224049062\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.794iT - 3T^{2} \)
7 \( 1 + 2.47iT - 7T^{2} \)
11 \( 1 + 2.29T + 11T^{2} \)
13 \( 1 + 3.84iT - 13T^{2} \)
17 \( 1 - 7.74iT - 17T^{2} \)
19 \( 1 - 2.29T + 19T^{2} \)
29 \( 1 + 5.28T + 29T^{2} \)
31 \( 1 + 6.40T + 31T^{2} \)
37 \( 1 + 8.56iT - 37T^{2} \)
41 \( 1 - 4.27T + 41T^{2} \)
43 \( 1 + 1.88iT - 43T^{2} \)
47 \( 1 + 12.3iT - 47T^{2} \)
53 \( 1 + 7.57iT - 53T^{2} \)
59 \( 1 + 6.07T + 59T^{2} \)
61 \( 1 + 0.635T + 61T^{2} \)
67 \( 1 + 11.1iT - 67T^{2} \)
71 \( 1 - 8.58T + 71T^{2} \)
73 \( 1 - 16.5iT - 73T^{2} \)
79 \( 1 + 0.335T + 79T^{2} \)
83 \( 1 - 15.1iT - 83T^{2} \)
89 \( 1 + 5.55T + 89T^{2} \)
97 \( 1 + 6.42iT - 97T^{2} \)
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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

−7.85771601162635669259925981186, −7.38194831997918020558356173014, −6.73349600961429717590409612948, −5.75734435216587769276954573466, −5.21696187022453887547214136448, −3.92894107326787477531994120286, −3.71653108994701176815663904300, −2.32245702870253218013955924472, −1.44781982575579223765552867465, −0.33643821475716445205698980173, 1.38683995820136882419082971015, 2.45599784625811697042424767458, 3.19572340252795069062776733344, 4.28689583780416074852469145202, 4.92798650439327093141302973704, 5.53645013995108366539649750176, 6.44406010646610696266542791880, 7.36714936971756997892216518668, 7.68037289148687569276358432320, 8.932101716087053894914602593073

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