Properties

Label 2-4598-1.1-c1-0-110
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s + 4-s − 0.618·5-s − 2·6-s + 1.85·7-s + 8-s + 9-s − 0.618·10-s − 2·12-s − 6.47·13-s + 1.85·14-s + 1.23·15-s + 16-s + 5.61·17-s + 18-s + 19-s − 0.618·20-s − 3.70·21-s − 1.14·23-s − 2·24-s − 4.61·25-s − 6.47·26-s + 4·27-s + 1.85·28-s + 1.52·29-s + 1.23·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 0.5·4-s − 0.276·5-s − 0.816·6-s + 0.700·7-s + 0.353·8-s + 0.333·9-s − 0.195·10-s − 0.577·12-s − 1.79·13-s + 0.495·14-s + 0.319·15-s + 0.250·16-s + 1.36·17-s + 0.235·18-s + 0.229·19-s − 0.138·20-s − 0.809·21-s − 0.238·23-s − 0.408·24-s − 0.923·25-s − 1.26·26-s + 0.769·27-s + 0.350·28-s + 0.283·29-s + 0.225·30-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: \(1\)
Selberg data: \((2,\ 4598,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T + 3T^{2} \)
5 \( 1 + 0.618T + 5T^{2} \)
7 \( 1 - 1.85T + 7T^{2} \)
13 \( 1 + 6.47T + 13T^{2} \)
17 \( 1 - 5.61T + 17T^{2} \)
23 \( 1 + 1.14T + 23T^{2} \)
29 \( 1 - 1.52T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 9.23T + 41T^{2} \)
43 \( 1 - 2.09T + 43T^{2} \)
47 \( 1 - 3.61T + 47T^{2} \)
53 \( 1 + 9.23T + 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 - 7.23T + 67T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 + 9.70T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 - 6.76T + 89T^{2} \)
97 \( 1 + 4.76T + 97T^{2} \)
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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

−7.65352371865183146561316864072, −7.25990791767006919086354467654, −6.23980455174058285061701918720, −5.66772664389623827503423585486, −4.90592583676155938046473541599, −4.62349981674817245075266384114, −3.44853846450588363611909517548, −2.51323547474406339157027294835, −1.35322078436533336793995264758, 0, 1.35322078436533336793995264758, 2.51323547474406339157027294835, 3.44853846450588363611909517548, 4.62349981674817245075266384114, 4.90592583676155938046473541599, 5.66772664389623827503423585486, 6.23980455174058285061701918720, 7.25990791767006919086354467654, 7.65352371865183146561316864072

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