Properties

Label 2-4598-1.1-c1-0-49
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 0.639·3-s + 4-s − 1.63·5-s − 0.639·6-s + 1.54·7-s + 8-s − 2.59·9-s − 1.63·10-s − 0.639·12-s + 6.04·13-s + 1.54·14-s + 1.04·15-s + 16-s − 2.54·17-s − 2.59·18-s + 19-s − 1.63·20-s − 0.990·21-s − 7.04·23-s − 0.639·24-s − 2.31·25-s + 6.04·26-s + 3.57·27-s + 1.54·28-s + 1.95·29-s + 1.04·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.369·3-s + 0.5·4-s − 0.733·5-s − 0.260·6-s + 0.585·7-s + 0.353·8-s − 0.863·9-s − 0.518·10-s − 0.184·12-s + 1.67·13-s + 0.414·14-s + 0.270·15-s + 0.250·16-s − 0.618·17-s − 0.610·18-s + 0.229·19-s − 0.366·20-s − 0.216·21-s − 1.46·23-s − 0.130·24-s − 0.462·25-s + 1.18·26-s + 0.687·27-s + 0.292·28-s + 0.362·29-s + 0.191·30-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

\(L(1)\) \(\approx\) \(2.336815440\)
\(L(\frac12)\) \(\approx\) \(2.336815440\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.639T + 3T^{2} \)
5 \( 1 + 1.63T + 5T^{2} \)
7 \( 1 - 1.54T + 7T^{2} \)
13 \( 1 - 6.04T + 13T^{2} \)
17 \( 1 + 2.54T + 17T^{2} \)
23 \( 1 + 7.04T + 23T^{2} \)
29 \( 1 - 1.95T + 29T^{2} \)
31 \( 1 + 1.27T + 31T^{2} \)
37 \( 1 - 5.73T + 37T^{2} \)
41 \( 1 - 9.13T + 41T^{2} \)
43 \( 1 - 12.8T + 43T^{2} \)
47 \( 1 - 3.59T + 47T^{2} \)
53 \( 1 + 7.95T + 53T^{2} \)
59 \( 1 - 5.40T + 59T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 - 10.2T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + 1.45T + 73T^{2} \)
79 \( 1 + 1.13T + 79T^{2} \)
83 \( 1 + 4.36T + 83T^{2} \)
89 \( 1 + 5.85T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.174997101224785166456933763914, −7.69953472497707007472632000658, −6.66715903923277840474754632546, −5.89461595572480936415458167533, −5.60262149052477330050583802751, −4.30777053859391735256563633832, −4.08004915013758772737223039029, −3.05583947662536031124272865837, −2.05656759438047773569047208272, −0.78044104114830551867073652068, 0.78044104114830551867073652068, 2.05656759438047773569047208272, 3.05583947662536031124272865837, 4.08004915013758772737223039029, 4.30777053859391735256563633832, 5.60262149052477330050583802751, 5.89461595572480936415458167533, 6.66715903923277840474754632546, 7.69953472497707007472632000658, 8.174997101224785166456933763914

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