Properties

Label 2-4600-1.1-c1-0-7
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  − 0.724·3-s − 2.33·7-s − 2.47·9-s − 2.62·11-s − 4.29·13-s + 6.85·17-s + 2.87·19-s + 1.69·21-s + 23-s + 3.96·27-s − 5.03·29-s − 7.31·31-s + 1.90·33-s − 9.24·37-s + 3.11·39-s − 6.95·41-s + 7.01·43-s + 4.74·47-s − 1.53·49-s − 4.96·51-s − 12.3·53-s − 2.08·57-s − 2.14·59-s − 2.30·61-s + 5.78·63-s − 1.98·67-s − 0.724·69-s + ⋯
L(s)  = 1  − 0.418·3-s − 0.883·7-s − 0.824·9-s − 0.792·11-s − 1.19·13-s + 1.66·17-s + 0.659·19-s + 0.369·21-s + 0.208·23-s + 0.763·27-s − 0.935·29-s − 1.31·31-s + 0.331·33-s − 1.51·37-s + 0.498·39-s − 1.08·41-s + 1.06·43-s + 0.692·47-s − 0.218·49-s − 0.695·51-s − 1.70·53-s − 0.275·57-s − 0.278·59-s − 0.295·61-s + 0.729·63-s − 0.242·67-s − 0.0872·69-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

\(L(1)\) \(\approx\) \(0.7134021293\)
\(L(\frac12)\) \(\approx\) \(0.7134021293\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.724T + 3T^{2} \)
7 \( 1 + 2.33T + 7T^{2} \)
11 \( 1 + 2.62T + 11T^{2} \)
13 \( 1 + 4.29T + 13T^{2} \)
17 \( 1 - 6.85T + 17T^{2} \)
19 \( 1 - 2.87T + 19T^{2} \)
29 \( 1 + 5.03T + 29T^{2} \)
31 \( 1 + 7.31T + 31T^{2} \)
37 \( 1 + 9.24T + 37T^{2} \)
41 \( 1 + 6.95T + 41T^{2} \)
43 \( 1 - 7.01T + 43T^{2} \)
47 \( 1 - 4.74T + 47T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 + 2.14T + 59T^{2} \)
61 \( 1 + 2.30T + 61T^{2} \)
67 \( 1 + 1.98T + 67T^{2} \)
71 \( 1 + 6.87T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 - 16.6T + 79T^{2} \)
83 \( 1 - 9.09T + 83T^{2} \)
89 \( 1 - 0.676T + 89T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−8.163089283293406864091086539550, −7.54904241192281861019362079074, −6.95542032722021814085702862925, −5.95148891289056340753144287361, −5.40568885061790263150882842944, −4.92107632250966524180419680410, −3.43420401595763752559117758396, −3.12948556148899701169402683173, −1.98884649052487709158668838898, −0.45389790267112783386339757269, 0.45389790267112783386339757269, 1.98884649052487709158668838898, 3.12948556148899701169402683173, 3.43420401595763752559117758396, 4.92107632250966524180419680410, 5.40568885061790263150882842944, 5.95148891289056340753144287361, 6.95542032722021814085702862925, 7.54904241192281861019362079074, 8.163089283293406864091086539550

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