Properties

Label 2-4598-1.1-c1-0-68
Degree $2$
Conductor $4598$
Sign $1$
Analytic cond. $36.7152$
Root an. cond. $6.05930$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3·3-s + 4-s − 3·6-s − 8-s + 6·9-s + 3·12-s − 7·13-s + 16-s + 7·17-s − 6·18-s − 19-s + 8·23-s − 3·24-s − 5·25-s + 7·26-s + 9·27-s + 9·29-s + 2·31-s − 32-s − 7·34-s + 6·36-s + 3·37-s + 38-s − 21·39-s − 10·41-s + 10·43-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.353·8-s + 2·9-s + 0.866·12-s − 1.94·13-s + 1/4·16-s + 1.69·17-s − 1.41·18-s − 0.229·19-s + 1.66·23-s − 0.612·24-s − 25-s + 1.37·26-s + 1.73·27-s + 1.67·29-s + 0.359·31-s − 0.176·32-s − 1.20·34-s + 36-s + 0.493·37-s + 0.162·38-s − 3.36·39-s − 1.56·41-s + 1.52·43-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: yes
Arithmetic: yes
Character: $\chi_{4598} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4598,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.770941468\)
\(L(\frac12)\) \(\approx\) \(2.770941468\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 + 7 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.343784787049288250151561526278, −7.59428648141978179934030198035, −7.41444755543721562166042908750, −6.50018362190960653186803724107, −5.25524637908827971867499931624, −4.48515576886248486520230602491, −3.34455255488757409394848077156, −2.81304263164298436582493772901, −2.10444152410754807673919512636, −0.971511438944262836171635213820, 0.971511438944262836171635213820, 2.10444152410754807673919512636, 2.81304263164298436582493772901, 3.34455255488757409394848077156, 4.48515576886248486520230602491, 5.25524637908827971867499931624, 6.50018362190960653186803724107, 7.41444755543721562166042908750, 7.59428648141978179934030198035, 8.343784787049288250151561526278

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