Properties

Label 2-4598-1.1-c1-0-100
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 + 3.01·3-s + 4-s − 2.00·5-s + 3.01·6-s + 2.42·7-s + 8-s + 6.08·9-s − 2.00·10-s + 3.01·12-s + 3.03·13-s + 2.42·14-s − 6.05·15-s + 16-s − 4.31·17-s + 6.08·18-s − 19-s − 2.00·20-s + 7.32·21-s − 2.27·23-s + 3.01·24-s − 0.960·25-s + 3.03·26-s + 9.30·27-s + 2.42·28-s + 8.81·29-s − 6.05·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.74·3-s + 0.5·4-s − 0.898·5-s + 1.23·6-s + 0.917·7-s + 0.353·8-s + 2.02·9-s − 0.635·10-s + 0.870·12-s + 0.841·13-s + 0.649·14-s − 1.56·15-s + 0.250·16-s − 1.04·17-s + 1.43·18-s − 0.229·19-s − 0.449·20-s + 1.59·21-s − 0.475·23-s + 0.615·24-s − 0.192·25-s + 0.595·26-s + 1.79·27-s + 0.458·28-s + 1.63·29-s − 1.10·30-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

\(L(1)\) \(\approx\) \(5.943420864\)
\(L(\frac12)\) \(\approx\) \(5.943420864\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.01T + 3T^{2} \)
5 \( 1 + 2.00T + 5T^{2} \)
7 \( 1 - 2.42T + 7T^{2} \)
13 \( 1 - 3.03T + 13T^{2} \)
17 \( 1 + 4.31T + 17T^{2} \)
23 \( 1 + 2.27T + 23T^{2} \)
29 \( 1 - 8.81T + 29T^{2} \)
31 \( 1 - 9.29T + 31T^{2} \)
37 \( 1 - 1.06T + 37T^{2} \)
41 \( 1 + 5.15T + 41T^{2} \)
43 \( 1 - 11.9T + 43T^{2} \)
47 \( 1 + 9.00T + 47T^{2} \)
53 \( 1 - 5.09T + 53T^{2} \)
59 \( 1 + 4.79T + 59T^{2} \)
61 \( 1 - 9.26T + 61T^{2} \)
67 \( 1 - 9.19T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 - 8.93T + 73T^{2} \)
79 \( 1 - 6.52T + 79T^{2} \)
83 \( 1 + 4.09T + 83T^{2} \)
89 \( 1 - 6.04T + 89T^{2} \)
97 \( 1 + 9.42T + 97T^{2} \)
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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

−8.129573974675886913830384368675, −7.992238453080517729671627145081, −6.93977922763763840557724214263, −6.34402621930353220810884696346, −4.99587261525464833915413353592, −4.22472709748023381537523683976, −3.93529574935267888904175509506, −2.92590346686923999020642995838, −2.28045417514315707256469793049, −1.25109273387972170750529827206, 1.25109273387972170750529827206, 2.28045417514315707256469793049, 2.92590346686923999020642995838, 3.93529574935267888904175509506, 4.22472709748023381537523683976, 4.99587261525464833915413353592, 6.34402621930353220810884696346, 6.93977922763763840557724214263, 7.992238453080517729671627145081, 8.129573974675886913830384368675

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