Properties

Label 2-4598-1.1-c1-0-2
Degree $2$
Conductor $4598$
Sign $1$
Analytic cond. $36.7152$
Root an. cond. $6.05930$
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-s − 0.103·3-s + 4-s − 3.91·5-s − 0.103·6-s − 4.45·7-s + 8-s − 2.98·9-s − 3.91·10-s − 0.103·12-s − 4.23·13-s − 4.45·14-s + 0.404·15-s + 16-s − 6.60·17-s − 2.98·18-s − 19-s − 3.91·20-s + 0.460·21-s − 1.59·23-s − 0.103·24-s + 10.3·25-s − 4.23·26-s + 0.618·27-s − 4.45·28-s − 6.26·29-s + 0.404·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.0596·3-s + 0.5·4-s − 1.75·5-s − 0.0421·6-s − 1.68·7-s + 0.353·8-s − 0.996·9-s − 1.23·10-s − 0.0298·12-s − 1.17·13-s − 1.19·14-s + 0.104·15-s + 0.250·16-s − 1.60·17-s − 0.704·18-s − 0.229·19-s − 0.876·20-s + 0.100·21-s − 0.332·23-s − 0.0210·24-s + 2.07·25-s − 0.829·26-s + 0.119·27-s − 0.842·28-s − 1.16·29-s + 0.0738·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4598 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4598\)    =    \(2 \cdot 11^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(36.7152\)
Root analytic conductor: \(6.05930\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4598} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4598,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1821466062\)
\(L(\frac12)\) \(\approx\) \(0.1821466062\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
11 \( 1 \)
19 \( 1 + T \)
good3 \( 1 + 0.103T + 3T^{2} \)
5 \( 1 + 3.91T + 5T^{2} \)
7 \( 1 + 4.45T + 7T^{2} \)
13 \( 1 + 4.23T + 13T^{2} \)
17 \( 1 + 6.60T + 17T^{2} \)
23 \( 1 + 1.59T + 23T^{2} \)
29 \( 1 + 6.26T + 29T^{2} \)
31 \( 1 + 6.39T + 31T^{2} \)
37 \( 1 - 8.97T + 37T^{2} \)
41 \( 1 - 1.91T + 41T^{2} \)
43 \( 1 - 9.24T + 43T^{2} \)
47 \( 1 + 4.59T + 47T^{2} \)
53 \( 1 + 4.30T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 + 0.325T + 61T^{2} \)
67 \( 1 - 8.88T + 67T^{2} \)
71 \( 1 + 2.66T + 71T^{2} \)
73 \( 1 + 6.64T + 73T^{2} \)
79 \( 1 + 2.13T + 79T^{2} \)
83 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 - 8.88T + 89T^{2} \)
97 \( 1 + 11.0T + 97T^{2} \)
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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

−8.082989538004123905823758026219, −7.50378189513499113670894991034, −6.80947565741690620563403337150, −6.22696476114278593384842614979, −5.33129081503094203662683117067, −4.32678631419165742380040842214, −3.87820469226525044969417923817, −3.01611749690718161917513252732, −2.44999987509506693875100040570, −0.19344243548290849982379309018, 0.19344243548290849982379309018, 2.44999987509506693875100040570, 3.01611749690718161917513252732, 3.87820469226525044969417923817, 4.32678631419165742380040842214, 5.33129081503094203662683117067, 6.22696476114278593384842614979, 6.80947565741690620563403337150, 7.50378189513499113670894991034, 8.082989538004123905823758026219

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