Properties

Label 2-4598-1.1-c1-0-14
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 + 2·3-s + 4-s − 2·5-s − 2·6-s − 4·7-s − 8-s + 9-s + 2·10-s + 2·12-s − 6·13-s + 4·14-s − 4·15-s + 16-s − 4·17-s − 18-s + 19-s − 2·20-s − 8·21-s + 8·23-s − 2·24-s − 25-s + 6·26-s − 4·27-s − 4·28-s + 6·29-s + 4·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.894·5-s − 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.577·12-s − 1.66·13-s + 1.06·14-s − 1.03·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.229·19-s − 0.447·20-s − 1.74·21-s + 1.66·23-s − 0.408·24-s − 1/5·25-s + 1.17·26-s − 0.769·27-s − 0.755·28-s + 1.11·29-s + 0.730·30-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: 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

\(L(1)\) \(\approx\) \(0.7322467590\)
\(L(\frac12)\) \(\approx\) \(0.7322467590\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−8.412964196449622001923808778207, −7.60084725755855945455554399752, −7.13443000516846209502549001680, −6.57364668896873915117551213392, −5.40195646820002768106416449469, −4.34138832625365150953948773707, −3.43453029260484114792237858781, −2.86828736843541374000917924324, −2.20839778779771438700510412326, −0.46222343669580291773504214779, 0.46222343669580291773504214779, 2.20839778779771438700510412326, 2.86828736843541374000917924324, 3.43453029260484114792237858781, 4.34138832625365150953948773707, 5.40195646820002768106416449469, 6.57364668896873915117551213392, 7.13443000516846209502549001680, 7.60084725755855945455554399752, 8.412964196449622001923808778207

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