Properties

Label 2-4598-1.1-c1-0-19
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 + 0.669·3-s + 4-s − 3.56·5-s − 0.669·6-s + 0.507·7-s − 8-s − 2.55·9-s + 3.56·10-s + 0.669·12-s + 6.56·13-s − 0.507·14-s − 2.38·15-s + 16-s − 4.23·17-s + 2.55·18-s + 19-s − 3.56·20-s + 0.339·21-s + 0.177·23-s − 0.669·24-s + 7.71·25-s − 6.56·26-s − 3.71·27-s + 0.507·28-s − 7.08·29-s + 2.38·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.386·3-s + 0.5·4-s − 1.59·5-s − 0.273·6-s + 0.191·7-s − 0.353·8-s − 0.850·9-s + 1.12·10-s + 0.193·12-s + 1.82·13-s − 0.135·14-s − 0.616·15-s + 0.250·16-s − 1.02·17-s + 0.601·18-s + 0.229·19-s − 0.797·20-s + 0.0741·21-s + 0.0369·23-s − 0.136·24-s + 1.54·25-s − 1.28·26-s − 0.715·27-s + 0.0958·28-s − 1.31·29-s + 0.436·30-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

\(L(1)\) \(\approx\) \(0.8237616148\)
\(L(\frac12)\) \(\approx\) \(0.8237616148\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.669T + 3T^{2} \)
5 \( 1 + 3.56T + 5T^{2} \)
7 \( 1 - 0.507T + 7T^{2} \)
13 \( 1 - 6.56T + 13T^{2} \)
17 \( 1 + 4.23T + 17T^{2} \)
23 \( 1 - 0.177T + 23T^{2} \)
29 \( 1 + 7.08T + 29T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 + 7.45T + 37T^{2} \)
41 \( 1 + 2.71T + 41T^{2} \)
43 \( 1 + 3.21T + 43T^{2} \)
47 \( 1 + 3.13T + 47T^{2} \)
53 \( 1 - 2.54T + 53T^{2} \)
59 \( 1 - 5.57T + 59T^{2} \)
61 \( 1 + 0.985T + 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 + 8.55T + 71T^{2} \)
73 \( 1 + 1.10T + 73T^{2} \)
79 \( 1 + 9.76T + 79T^{2} \)
83 \( 1 - 8.19T + 83T^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 + 8.82T + 97T^{2} \)
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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

−8.333195484510422725895818258836, −7.972713574921236502788987836127, −6.99800261655238769809674879539, −6.41458045782135616995602481240, −5.44522768966150804007334306952, −4.36282948946776859234760809585, −3.59530387491869567977843992578, −3.05301460317335736327961977033, −1.79581628320343702783251276045, −0.54512057960230178295176708463, 0.54512057960230178295176708463, 1.79581628320343702783251276045, 3.05301460317335736327961977033, 3.59530387491869567977843992578, 4.36282948946776859234760809585, 5.44522768966150804007334306952, 6.41458045782135616995602481240, 6.99800261655238769809674879539, 7.972713574921236502788987836127, 8.333195484510422725895818258836

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