Properties

Label 2-4598-1.1-c1-0-11
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 − 2.23·3-s + 4-s + 0.618·5-s + 2.23·6-s − 2.85·7-s − 8-s + 2.00·9-s − 0.618·10-s − 2.23·12-s − 1.61·13-s + 2.85·14-s − 1.38·15-s + 16-s + 6.23·17-s − 2.00·18-s − 19-s + 0.618·20-s + 6.38·21-s − 0.236·23-s + 2.23·24-s − 4.61·25-s + 1.61·26-s + 2.23·27-s − 2.85·28-s + 10.4·29-s + 1.38·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.29·3-s + 0.5·4-s + 0.276·5-s + 0.912·6-s − 1.07·7-s − 0.353·8-s + 0.666·9-s − 0.195·10-s − 0.645·12-s − 0.448·13-s + 0.762·14-s − 0.356·15-s + 0.250·16-s + 1.51·17-s − 0.471·18-s − 0.229·19-s + 0.138·20-s + 1.39·21-s − 0.0492·23-s + 0.456·24-s − 0.923·25-s + 0.317·26-s + 0.430·27-s − 0.539·28-s + 1.94·29-s + 0.252·30-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

\(L(1)\) \(\approx\) \(0.4436135683\)
\(L(\frac12)\) \(\approx\) \(0.4436135683\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23T + 3T^{2} \)
5 \( 1 - 0.618T + 5T^{2} \)
7 \( 1 + 2.85T + 7T^{2} \)
13 \( 1 + 1.61T + 13T^{2} \)
17 \( 1 - 6.23T + 17T^{2} \)
23 \( 1 + 0.236T + 23T^{2} \)
29 \( 1 - 10.4T + 29T^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 + 7.85T + 37T^{2} \)
41 \( 1 + 4.09T + 41T^{2} \)
43 \( 1 + 3.70T + 43T^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 - 5.94T + 53T^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 - 4.38T + 61T^{2} \)
67 \( 1 + 6.09T + 67T^{2} \)
71 \( 1 + 3.76T + 71T^{2} \)
73 \( 1 + 1.32T + 73T^{2} \)
79 \( 1 + 3T + 79T^{2} \)
83 \( 1 + 1.09T + 83T^{2} \)
89 \( 1 + 9.18T + 89T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−8.384128100991849732609244420164, −7.36487406471447669889341876599, −6.91171087307511925945520357482, −6.03653597334224017526757929125, −5.70255533127782000756365915216, −4.87835054082976535854434014287, −3.66165213990602140085641009450, −2.84291072239016416493754864037, −1.59023624936314229582331173761, −0.44397210129866027016585612795, 0.44397210129866027016585612795, 1.59023624936314229582331173761, 2.84291072239016416493754864037, 3.66165213990602140085641009450, 4.87835054082976535854434014287, 5.70255533127782000756365915216, 6.03653597334224017526757929125, 6.91171087307511925945520357482, 7.36487406471447669889341876599, 8.384128100991849732609244420164

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