Properties

Label 2-4598-1.1-c1-0-9
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 − 3.19·3-s + 4-s − 4.19·5-s − 3.19·6-s + 1.32·7-s + 8-s + 7.20·9-s − 4.19·10-s − 3.19·12-s − 3.52·13-s + 1.32·14-s + 13.4·15-s + 16-s − 2.32·17-s + 7.20·18-s + 19-s − 4.19·20-s − 4.22·21-s + 2.52·23-s − 3.19·24-s + 12.5·25-s − 3.52·26-s − 13.4·27-s + 1.32·28-s − 10.4·29-s + 13.4·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.84·3-s + 0.5·4-s − 1.87·5-s − 1.30·6-s + 0.499·7-s + 0.353·8-s + 2.40·9-s − 1.32·10-s − 0.922·12-s − 0.978·13-s + 0.353·14-s + 3.45·15-s + 0.250·16-s − 0.563·17-s + 1.69·18-s + 0.229·19-s − 0.937·20-s − 0.922·21-s + 0.527·23-s − 0.652·24-s + 2.51·25-s − 0.691·26-s − 2.58·27-s + 0.249·28-s − 1.93·29-s + 2.44·30-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

\(L(1)\) \(\approx\) \(0.5471701826\)
\(L(\frac12)\) \(\approx\) \(0.5471701826\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.19T + 3T^{2} \)
5 \( 1 + 4.19T + 5T^{2} \)
7 \( 1 - 1.32T + 7T^{2} \)
13 \( 1 + 3.52T + 13T^{2} \)
17 \( 1 + 2.32T + 17T^{2} \)
23 \( 1 - 2.52T + 23T^{2} \)
29 \( 1 + 10.4T + 29T^{2} \)
31 \( 1 + 6.38T + 31T^{2} \)
37 \( 1 - 7.84T + 37T^{2} \)
41 \( 1 - 2.34T + 41T^{2} \)
43 \( 1 + 11.4T + 43T^{2} \)
47 \( 1 + 6.20T + 47T^{2} \)
53 \( 1 - 4.40T + 53T^{2} \)
59 \( 1 + 6.72T + 59T^{2} \)
61 \( 1 + 8.60T + 61T^{2} \)
67 \( 1 - 6.21T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 + 1.67T + 73T^{2} \)
79 \( 1 - 5.65T + 79T^{2} \)
83 \( 1 + 1.80T + 83T^{2} \)
89 \( 1 - 6.04T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−7.78072168267982225453536287917, −7.45937443523248626181029782756, −6.87038399836708409451634990687, −6.04052222900966930136549851581, −5.08379249782215090096281305567, −4.77589068985574244236515326314, −4.10337334805057890914148738095, −3.29391591670469208497471683845, −1.73277410303923656236999009676, −0.40387372729302096568988663116, 0.40387372729302096568988663116, 1.73277410303923656236999009676, 3.29391591670469208497471683845, 4.10337334805057890914148738095, 4.77589068985574244236515326314, 5.08379249782215090096281305567, 6.04052222900966930136549851581, 6.87038399836708409451634990687, 7.45937443523248626181029782756, 7.78072168267982225453536287917

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