Properties

Label 2-4600-1.1-c1-0-33
Degree $2$
Conductor $4600$
Sign $1$
Analytic cond. $36.7311$
Root an. cond. $6.06062$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.83·3-s − 3.97·7-s + 0.384·9-s + 2.10·11-s + 5.35·13-s + 1.29·17-s + 2.10·19-s − 7.30·21-s − 23-s − 4.81·27-s + 6.03·29-s − 8.32·31-s + 3.86·33-s + 5.10·37-s + 9.85·39-s − 8.33·41-s + 7.78·43-s + 11.3·47-s + 8.78·49-s + 2.38·51-s − 0.573·53-s + 3.86·57-s − 9.17·59-s + 13.5·61-s − 1.52·63-s − 15.8·67-s − 1.83·69-s + ⋯
L(s)  = 1  + 1.06·3-s − 1.50·7-s + 0.128·9-s + 0.634·11-s + 1.48·13-s + 0.314·17-s + 0.482·19-s − 1.59·21-s − 0.208·23-s − 0.926·27-s + 1.11·29-s − 1.49·31-s + 0.673·33-s + 0.838·37-s + 1.57·39-s − 1.30·41-s + 1.18·43-s + 1.66·47-s + 1.25·49-s + 0.333·51-s − 0.0787·53-s + 0.512·57-s − 1.19·59-s + 1.73·61-s − 0.192·63-s − 1.93·67-s − 0.221·69-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(2.551332857\)
\(L(\frac12)\) \(\approx\) \(2.551332857\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good3 \( 1 - 1.83T + 3T^{2} \)
7 \( 1 + 3.97T + 7T^{2} \)
11 \( 1 - 2.10T + 11T^{2} \)
13 \( 1 - 5.35T + 13T^{2} \)
17 \( 1 - 1.29T + 17T^{2} \)
19 \( 1 - 2.10T + 19T^{2} \)
29 \( 1 - 6.03T + 29T^{2} \)
31 \( 1 + 8.32T + 31T^{2} \)
37 \( 1 - 5.10T + 37T^{2} \)
41 \( 1 + 8.33T + 41T^{2} \)
43 \( 1 - 7.78T + 43T^{2} \)
47 \( 1 - 11.3T + 47T^{2} \)
53 \( 1 + 0.573T + 53T^{2} \)
59 \( 1 + 9.17T + 59T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 + 15.8T + 67T^{2} \)
71 \( 1 - 14.9T + 71T^{2} \)
73 \( 1 - 8.36T + 73T^{2} \)
79 \( 1 + 9.41T + 79T^{2} \)
83 \( 1 - 1.51T + 83T^{2} \)
89 \( 1 - 10.5T + 89T^{2} \)
97 \( 1 + 0.337T + 97T^{2} \)
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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

−8.496986063375097438588073201233, −7.65774249854845684838945129821, −6.89997405445086573112836953923, −6.14470501272361008750308374417, −5.65065003750068433573606536440, −4.22688832558102923691103565392, −3.48593461368859395516351503268, −3.16981120625340703592729183770, −2.10503048406690081825881614053, −0.848774523513021746028498612602, 0.848774523513021746028498612602, 2.10503048406690081825881614053, 3.16981120625340703592729183770, 3.48593461368859395516351503268, 4.22688832558102923691103565392, 5.65065003750068433573606536440, 6.14470501272361008750308374417, 6.89997405445086573112836953923, 7.65774249854845684838945129821, 8.496986063375097438588073201233

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