Properties

Label 2-4598-1.1-c1-0-140
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 + 2.10·3-s + 4-s + 2.46·5-s + 2.10·6-s + 3.09·7-s + 8-s + 1.41·9-s + 2.46·10-s + 2.10·12-s + 3.97·13-s + 3.09·14-s + 5.16·15-s + 16-s + 2.14·17-s + 1.41·18-s + 19-s + 2.46·20-s + 6.51·21-s − 2.55·23-s + 2.10·24-s + 1.05·25-s + 3.97·26-s − 3.33·27-s + 3.09·28-s − 6.50·29-s + 5.16·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.21·3-s + 0.5·4-s + 1.10·5-s + 0.857·6-s + 1.17·7-s + 0.353·8-s + 0.471·9-s + 0.777·10-s + 0.606·12-s + 1.10·13-s + 0.828·14-s + 1.33·15-s + 0.250·16-s + 0.520·17-s + 0.333·18-s + 0.229·19-s + 0.550·20-s + 1.42·21-s − 0.533·23-s + 0.428·24-s + 0.210·25-s + 0.780·26-s − 0.641·27-s + 0.585·28-s − 1.20·29-s + 0.943·30-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

\(L(1)\) \(\approx\) \(6.901897205\)
\(L(\frac12)\) \(\approx\) \(6.901897205\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.10T + 3T^{2} \)
5 \( 1 - 2.46T + 5T^{2} \)
7 \( 1 - 3.09T + 7T^{2} \)
13 \( 1 - 3.97T + 13T^{2} \)
17 \( 1 - 2.14T + 17T^{2} \)
23 \( 1 + 2.55T + 23T^{2} \)
29 \( 1 + 6.50T + 29T^{2} \)
31 \( 1 - 1.48T + 31T^{2} \)
37 \( 1 + 1.74T + 37T^{2} \)
41 \( 1 + 5.48T + 41T^{2} \)
43 \( 1 + 4.59T + 43T^{2} \)
47 \( 1 + 3.40T + 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + 6.65T + 59T^{2} \)
61 \( 1 - 6.34T + 61T^{2} \)
67 \( 1 - 14.1T + 67T^{2} \)
71 \( 1 + 15.9T + 71T^{2} \)
73 \( 1 - 3.17T + 73T^{2} \)
79 \( 1 + 7.25T + 79T^{2} \)
83 \( 1 + 11.2T + 83T^{2} \)
89 \( 1 + 4.63T + 89T^{2} \)
97 \( 1 + 5.47T + 97T^{2} \)
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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

−8.237847919962720763508139663562, −7.79988593305413301805084337576, −6.84414443540140530781052228949, −5.90591304512157596283846012617, −5.45656698477550974392222984191, −4.53835474699350940327572077776, −3.63477536425187220689369549234, −2.98376173536696584685277487745, −1.84388136967659744760366264769, −1.62369424655647907428268238721, 1.62369424655647907428268238721, 1.84388136967659744760366264769, 2.98376173536696584685277487745, 3.63477536425187220689369549234, 4.53835474699350940327572077776, 5.45656698477550974392222984191, 5.90591304512157596283846012617, 6.84414443540140530781052228949, 7.79988593305413301805084337576, 8.237847919962720763508139663562

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