Properties

Label 2-4600-1.1-c1-0-62
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  + 3.36·3-s + 1.90·7-s + 8.28·9-s − 5.48·11-s + 1.04·13-s + 6.74·17-s − 1.55·19-s + 6.40·21-s + 23-s + 17.7·27-s − 3.38·29-s + 10.9·31-s − 18.4·33-s − 5.26·37-s + 3.52·39-s + 6.09·41-s − 0.403·47-s − 3.36·49-s + 22.6·51-s − 5.88·53-s − 5.20·57-s + 9.60·59-s − 7.09·61-s + 15.8·63-s − 13.7·67-s + 3.36·69-s + 0.478·71-s + ⋯
L(s)  = 1  + 1.93·3-s + 0.720·7-s + 2.76·9-s − 1.65·11-s + 0.291·13-s + 1.63·17-s − 0.355·19-s + 1.39·21-s + 0.208·23-s + 3.42·27-s − 0.628·29-s + 1.96·31-s − 3.20·33-s − 0.865·37-s + 0.564·39-s + 0.952·41-s − 0.0589·47-s − 0.480·49-s + 3.17·51-s − 0.808·53-s − 0.690·57-s + 1.24·59-s − 0.908·61-s + 1.99·63-s − 1.68·67-s + 0.404·69-s + 0.0568·71-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

\(L(1)\) \(\approx\) \(4.614315000\)
\(L(\frac12)\) \(\approx\) \(4.614315000\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.36T + 3T^{2} \)
7 \( 1 - 1.90T + 7T^{2} \)
11 \( 1 + 5.48T + 11T^{2} \)
13 \( 1 - 1.04T + 13T^{2} \)
17 \( 1 - 6.74T + 17T^{2} \)
19 \( 1 + 1.55T + 19T^{2} \)
29 \( 1 + 3.38T + 29T^{2} \)
31 \( 1 - 10.9T + 31T^{2} \)
37 \( 1 + 5.26T + 37T^{2} \)
41 \( 1 - 6.09T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 0.403T + 47T^{2} \)
53 \( 1 + 5.88T + 53T^{2} \)
59 \( 1 - 9.60T + 59T^{2} \)
61 \( 1 + 7.09T + 61T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 - 0.478T + 71T^{2} \)
73 \( 1 + 2.40T + 73T^{2} \)
79 \( 1 + 4.24T + 79T^{2} \)
83 \( 1 - 11.2T + 83T^{2} \)
89 \( 1 - 4.90T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−8.270235227702952501533373830978, −7.68011408261211629488701087702, −7.46161830427824289372200037329, −6.21047235079983200000371240760, −5.12073815423603242315876814382, −4.51377277356136107735259255591, −3.48787491713065917975124773687, −2.90444314444126982771166507205, −2.15256287826063255202293680614, −1.19456314813965826176025391869, 1.19456314813965826176025391869, 2.15256287826063255202293680614, 2.90444314444126982771166507205, 3.48787491713065917975124773687, 4.51377277356136107735259255591, 5.12073815423603242315876814382, 6.21047235079983200000371240760, 7.46161830427824289372200037329, 7.68011408261211629488701087702, 8.270235227702952501533373830978

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