Properties

Label 2-4600-1.1-c1-0-1
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  − 2.66·3-s − 3.66·7-s + 4.12·9-s − 1.21·11-s − 2.21·13-s − 1.21·17-s − 2.57·19-s + 9.79·21-s − 23-s − 3.00·27-s + 1.45·29-s − 6.46·31-s + 3.24·33-s − 4·37-s + 5.90·39-s − 10.9·41-s − 6.90·43-s − 5.45·47-s + 6.46·49-s + 3.24·51-s − 3.81·53-s + 6.88·57-s − 4.24·59-s + 6.78·61-s − 15.1·63-s − 12.8·67-s + 2.66·69-s + ⋯
L(s)  = 1  − 1.54·3-s − 1.38·7-s + 1.37·9-s − 0.366·11-s − 0.614·13-s − 0.294·17-s − 0.591·19-s + 2.13·21-s − 0.208·23-s − 0.577·27-s + 0.270·29-s − 1.16·31-s + 0.564·33-s − 0.657·37-s + 0.946·39-s − 1.70·41-s − 1.05·43-s − 0.795·47-s + 0.923·49-s + 0.453·51-s − 0.524·53-s + 0.911·57-s − 0.553·59-s + 0.868·61-s − 1.90·63-s − 1.57·67-s + 0.321·69-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

\(L(1)\) \(\approx\) \(0.1474641639\)
\(L(\frac12)\) \(\approx\) \(0.1474641639\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.66T + 3T^{2} \)
7 \( 1 + 3.66T + 7T^{2} \)
11 \( 1 + 1.21T + 11T^{2} \)
13 \( 1 + 2.21T + 13T^{2} \)
17 \( 1 + 1.21T + 17T^{2} \)
19 \( 1 + 2.57T + 19T^{2} \)
29 \( 1 - 1.45T + 29T^{2} \)
31 \( 1 + 6.46T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 + 6.90T + 43T^{2} \)
47 \( 1 + 5.45T + 47T^{2} \)
53 \( 1 + 3.81T + 53T^{2} \)
59 \( 1 + 4.24T + 59T^{2} \)
61 \( 1 - 6.78T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 + 6.91T + 71T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 + 1.69T + 97T^{2} \)
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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

−8.289851458575642254962538619370, −7.26874487083893868819458253128, −6.61867034052630599078262658942, −6.28969265132007406947399480446, −5.35764232093868238499225341117, −4.89399109171753279059364371614, −3.85335546859498625934956509892, −2.98551025108616736907057122416, −1.75932585083826764026719011128, −0.22160359030539828221124059868, 0.22160359030539828221124059868, 1.75932585083826764026719011128, 2.98551025108616736907057122416, 3.85335546859498625934956509892, 4.89399109171753279059364371614, 5.35764232093868238499225341117, 6.28969265132007406947399480446, 6.61867034052630599078262658942, 7.26874487083893868819458253128, 8.289851458575642254962538619370

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