Properties

Label 2-4598-1.1-c1-0-109
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 − 2.16·3-s + 4-s + 4.16·5-s + 2.16·6-s + 0.683·7-s − 8-s + 1.68·9-s − 4.16·10-s − 2.16·12-s − 2.68·13-s − 0.683·14-s − 9.01·15-s + 16-s − 3.36·17-s − 1.68·18-s + 19-s + 4.16·20-s − 1.48·21-s − 1.36·23-s + 2.16·24-s + 12.3·25-s + 2.68·26-s + 2.84·27-s + 0.683·28-s − 2.79·29-s + 9.01·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.24·3-s + 0.5·4-s + 1.86·5-s + 0.883·6-s + 0.258·7-s − 0.353·8-s + 0.561·9-s − 1.31·10-s − 0.624·12-s − 0.744·13-s − 0.182·14-s − 2.32·15-s + 0.250·16-s − 0.816·17-s − 0.396·18-s + 0.229·19-s + 0.931·20-s − 0.323·21-s − 0.285·23-s + 0.441·24-s + 2.46·25-s + 0.526·26-s + 0.548·27-s + 0.129·28-s − 0.519·29-s + 1.64·30-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 \( 1 + 2.16T + 3T^{2} \)
5 \( 1 - 4.16T + 5T^{2} \)
7 \( 1 - 0.683T + 7T^{2} \)
13 \( 1 + 2.68T + 13T^{2} \)
17 \( 1 + 3.36T + 17T^{2} \)
23 \( 1 + 1.36T + 23T^{2} \)
29 \( 1 + 2.79T + 29T^{2} \)
31 \( 1 - 5.01T + 31T^{2} \)
37 \( 1 + 11.6T + 37T^{2} \)
41 \( 1 - 7.01T + 41T^{2} \)
43 \( 1 + 7.86T + 43T^{2} \)
47 \( 1 - 2.96T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + 9.92T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 0.683T + 67T^{2} \)
71 \( 1 - 3.53T + 71T^{2} \)
73 \( 1 + 0.407T + 73T^{2} \)
79 \( 1 - 7.28T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−8.006620409487818135389735778530, −6.89399418351089983991923732378, −6.53130714461499327761721513249, −5.83429289429203503319609857790, −5.22184170617061459442466833921, −4.64267974149214275071590775314, −2.99020902202705458000786845944, −2.07115015555569128349213722405, −1.34966991693543028260195209770, 0, 1.34966991693543028260195209770, 2.07115015555569128349213722405, 2.99020902202705458000786845944, 4.64267974149214275071590775314, 5.22184170617061459442466833921, 5.83429289429203503319609857790, 6.53130714461499327761721513249, 6.89399418351089983991923732378, 8.006620409487818135389735778530

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