Properties

Label 2-4598-1.1-c1-0-74
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 − 2.48·3-s + 4-s + 3.25·5-s − 2.48·6-s + 4.58·7-s + 8-s + 3.19·9-s + 3.25·10-s − 2.48·12-s − 4.82·13-s + 4.58·14-s − 8.10·15-s + 16-s + 1.77·17-s + 3.19·18-s − 19-s + 3.25·20-s − 11.4·21-s + 6.51·23-s − 2.48·24-s + 5.62·25-s − 4.82·26-s − 0.475·27-s + 4.58·28-s − 0.0103·29-s − 8.10·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.43·3-s + 0.5·4-s + 1.45·5-s − 1.01·6-s + 1.73·7-s + 0.353·8-s + 1.06·9-s + 1.03·10-s − 0.718·12-s − 1.33·13-s + 1.22·14-s − 2.09·15-s + 0.250·16-s + 0.431·17-s + 0.752·18-s − 0.229·19-s + 0.728·20-s − 2.49·21-s + 1.35·23-s − 0.507·24-s + 1.12·25-s − 0.945·26-s − 0.0915·27-s + 0.867·28-s − 0.00192·29-s − 1.48·30-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

\(L(1)\) \(\approx\) \(3.132182184\)
\(L(\frac12)\) \(\approx\) \(3.132182184\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.48T + 3T^{2} \)
5 \( 1 - 3.25T + 5T^{2} \)
7 \( 1 - 4.58T + 7T^{2} \)
13 \( 1 + 4.82T + 13T^{2} \)
17 \( 1 - 1.77T + 17T^{2} \)
23 \( 1 - 6.51T + 23T^{2} \)
29 \( 1 + 0.0103T + 29T^{2} \)
31 \( 1 + 3.91T + 31T^{2} \)
37 \( 1 - 0.688T + 37T^{2} \)
41 \( 1 - 3.27T + 41T^{2} \)
43 \( 1 - 1.38T + 43T^{2} \)
47 \( 1 + 6.30T + 47T^{2} \)
53 \( 1 - 14.4T + 53T^{2} \)
59 \( 1 + 9.20T + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 - 5.08T + 67T^{2} \)
71 \( 1 - 9.43T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 - 2.32T + 83T^{2} \)
89 \( 1 + 6.04T + 89T^{2} \)
97 \( 1 + 19.4T + 97T^{2} \)
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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

−8.163574105436910951205003566822, −7.23794928304396808861642922251, −6.73590648963692249222714327689, −5.76530959969015483130498025840, −5.28204907352464111183033751922, −5.03743233795261113398388662170, −4.23882740458221172239061403806, −2.66315133853890061524559869592, −1.89038532600464755117081585907, −1.02085640991249966557719034849, 1.02085640991249966557719034849, 1.89038532600464755117081585907, 2.66315133853890061524559869592, 4.23882740458221172239061403806, 5.03743233795261113398388662170, 5.28204907352464111183033751922, 5.76530959969015483130498025840, 6.73590648963692249222714327689, 7.23794928304396808861642922251, 8.163574105436910951205003566822

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