Properties

Label 2-4600-1.1-c1-0-41
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  + 1.33·3-s + 3.16·7-s − 1.21·9-s − 0.0955·11-s − 1.44·13-s + 2.29·17-s + 7.00·19-s + 4.23·21-s + 23-s − 5.63·27-s + 5.39·29-s − 0.584·31-s − 0.127·33-s + 9.29·37-s − 1.92·39-s − 2.86·41-s − 9.50·43-s − 7.09·47-s + 3.03·49-s + 3.07·51-s + 7.73·53-s + 9.36·57-s + 13.6·59-s + 0.234·61-s − 3.84·63-s − 7.49·67-s + 1.33·69-s + ⋯
L(s)  = 1  + 0.771·3-s + 1.19·7-s − 0.404·9-s − 0.0288·11-s − 0.400·13-s + 0.557·17-s + 1.60·19-s + 0.924·21-s + 0.208·23-s − 1.08·27-s + 1.00·29-s − 0.104·31-s − 0.0222·33-s + 1.52·37-s − 0.308·39-s − 0.447·41-s − 1.44·43-s − 1.03·47-s + 0.433·49-s + 0.430·51-s + 1.06·53-s + 1.24·57-s + 1.77·59-s + 0.0300·61-s − 0.483·63-s − 0.915·67-s + 0.160·69-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

\(L(1)\) \(\approx\) \(3.026495341\)
\(L(\frac12)\) \(\approx\) \(3.026495341\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.33T + 3T^{2} \)
7 \( 1 - 3.16T + 7T^{2} \)
11 \( 1 + 0.0955T + 11T^{2} \)
13 \( 1 + 1.44T + 13T^{2} \)
17 \( 1 - 2.29T + 17T^{2} \)
19 \( 1 - 7.00T + 19T^{2} \)
29 \( 1 - 5.39T + 29T^{2} \)
31 \( 1 + 0.584T + 31T^{2} \)
37 \( 1 - 9.29T + 37T^{2} \)
41 \( 1 + 2.86T + 41T^{2} \)
43 \( 1 + 9.50T + 43T^{2} \)
47 \( 1 + 7.09T + 47T^{2} \)
53 \( 1 - 7.73T + 53T^{2} \)
59 \( 1 - 13.6T + 59T^{2} \)
61 \( 1 - 0.234T + 61T^{2} \)
67 \( 1 + 7.49T + 67T^{2} \)
71 \( 1 + 5.18T + 71T^{2} \)
73 \( 1 - 1.52T + 73T^{2} \)
79 \( 1 + 3.04T + 79T^{2} \)
83 \( 1 - 15.9T + 83T^{2} \)
89 \( 1 - 5.53T + 89T^{2} \)
97 \( 1 + 2.58T + 97T^{2} \)
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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

−8.286048518287117246155030208008, −7.74198139034687589994219498482, −7.15719616804051576864382442964, −6.05819600284670345490178127277, −5.21058343201742731354202949035, −4.73481987582282592891317122046, −3.58725029922366523302900208259, −2.91277295066977399845641036854, −2.01576515114647906952004263852, −0.976781587510391934631316268951, 0.976781587510391934631316268951, 2.01576515114647906952004263852, 2.91277295066977399845641036854, 3.58725029922366523302900208259, 4.73481987582282592891317122046, 5.21058343201742731354202949035, 6.05819600284670345490178127277, 7.15719616804051576864382442964, 7.74198139034687589994219498482, 8.286048518287117246155030208008

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