Properties

Label 2-4600-1.1-c1-0-13
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  − 0.794·3-s + 2.47·7-s − 2.36·9-s − 2.29·11-s − 3.84·13-s − 7.74·17-s − 2.29·19-s − 1.96·21-s − 23-s + 4.26·27-s + 5.28·29-s − 6.40·31-s + 1.82·33-s + 8.56·37-s + 3.05·39-s + 4.27·41-s − 1.88·43-s + 12.3·47-s − 0.889·49-s + 6.15·51-s − 7.57·53-s + 1.82·57-s + 6.07·59-s − 0.635·61-s − 5.85·63-s + 11.1·67-s + 0.794·69-s + ⋯
L(s)  = 1  − 0.458·3-s + 0.934·7-s − 0.789·9-s − 0.693·11-s − 1.06·13-s − 1.87·17-s − 0.527·19-s − 0.428·21-s − 0.208·23-s + 0.821·27-s + 0.981·29-s − 1.14·31-s + 0.318·33-s + 1.40·37-s + 0.488·39-s + 0.667·41-s − 0.288·43-s + 1.80·47-s − 0.127·49-s + 0.862·51-s − 1.04·53-s + 0.242·57-s + 0.790·59-s − 0.0813·61-s − 0.737·63-s + 1.36·67-s + 0.0956·69-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

\(L(1)\) \(\approx\) \(1.030344264\)
\(L(\frac12)\) \(\approx\) \(1.030344264\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.794T + 3T^{2} \)
7 \( 1 - 2.47T + 7T^{2} \)
11 \( 1 + 2.29T + 11T^{2} \)
13 \( 1 + 3.84T + 13T^{2} \)
17 \( 1 + 7.74T + 17T^{2} \)
19 \( 1 + 2.29T + 19T^{2} \)
29 \( 1 - 5.28T + 29T^{2} \)
31 \( 1 + 6.40T + 31T^{2} \)
37 \( 1 - 8.56T + 37T^{2} \)
41 \( 1 - 4.27T + 41T^{2} \)
43 \( 1 + 1.88T + 43T^{2} \)
47 \( 1 - 12.3T + 47T^{2} \)
53 \( 1 + 7.57T + 53T^{2} \)
59 \( 1 - 6.07T + 59T^{2} \)
61 \( 1 + 0.635T + 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 - 8.58T + 71T^{2} \)
73 \( 1 - 16.5T + 73T^{2} \)
79 \( 1 - 0.335T + 79T^{2} \)
83 \( 1 - 15.1T + 83T^{2} \)
89 \( 1 - 5.55T + 89T^{2} \)
97 \( 1 - 6.42T + 97T^{2} \)
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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

−8.233304456568766993319744945058, −7.71816327968451243182689461686, −6.78832685433913313051253475037, −6.15620447667118786163399060537, −5.18115716494846645001023078658, −4.82745333768920034223328865853, −3.97460475904813273782382496982, −2.54364560648639716520477317487, −2.19036557682398692104585216422, −0.54656808089251075008767709253, 0.54656808089251075008767709253, 2.19036557682398692104585216422, 2.54364560648639716520477317487, 3.97460475904813273782382496982, 4.82745333768920034223328865853, 5.18115716494846645001023078658, 6.15620447667118786163399060537, 6.78832685433913313051253475037, 7.71816327968451243182689461686, 8.233304456568766993319744945058

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