Properties

Label 2-4598-1.1-c1-0-135
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 − 1.36·3-s + 4-s + 0.363·5-s − 1.36·6-s + 2.14·7-s + 8-s − 1.14·9-s + 0.363·10-s − 1.36·12-s − 3.14·13-s + 2.14·14-s − 0.495·15-s + 16-s + 1.14·17-s − 1.14·18-s − 19-s + 0.363·20-s − 2.91·21-s − 2.14·23-s − 1.36·24-s − 4.86·25-s − 3.14·26-s + 5.64·27-s + 2.14·28-s − 1.77·29-s − 0.495·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.787·3-s + 0.5·4-s + 0.162·5-s − 0.556·6-s + 0.809·7-s + 0.353·8-s − 0.380·9-s + 0.114·10-s − 0.393·12-s − 0.871·13-s + 0.572·14-s − 0.127·15-s + 0.250·16-s + 0.276·17-s − 0.269·18-s − 0.229·19-s + 0.0812·20-s − 0.637·21-s − 0.446·23-s − 0.278·24-s − 0.973·25-s − 0.616·26-s + 1.08·27-s + 0.404·28-s − 0.330·29-s − 0.0904·30-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 \( 1 + 1.36T + 3T^{2} \)
5 \( 1 - 0.363T + 5T^{2} \)
7 \( 1 - 2.14T + 7T^{2} \)
13 \( 1 + 3.14T + 13T^{2} \)
17 \( 1 - 1.14T + 17T^{2} \)
23 \( 1 + 2.14T + 23T^{2} \)
29 \( 1 + 1.77T + 29T^{2} \)
31 \( 1 - 7.00T + 31T^{2} \)
37 \( 1 + 9.77T + 37T^{2} \)
41 \( 1 + 7.91T + 41T^{2} \)
43 \( 1 + 4.28T + 43T^{2} \)
47 \( 1 + 7.59T + 47T^{2} \)
53 \( 1 - 10.7T + 53T^{2} \)
59 \( 1 + 1.08T + 59T^{2} \)
61 \( 1 + 8.51T + 61T^{2} \)
67 \( 1 - 8.56T + 67T^{2} \)
71 \( 1 - 3.50T + 71T^{2} \)
73 \( 1 - 11.7T + 73T^{2} \)
79 \( 1 - 0.778T + 79T^{2} \)
83 \( 1 + 0.778T + 83T^{2} \)
89 \( 1 + 8.20T + 89T^{2} \)
97 \( 1 + 1.86T + 97T^{2} \)
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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

−8.025358327943759140158439447131, −6.94018233601734903851122908215, −6.45491054849334739547649896690, −5.45297460513828538521207937878, −5.20156982604335854050014148043, −4.41598122049440959941774048961, −3.44894263179504292792870571708, −2.42662095296569579447857421198, −1.53529571695812710730601693159, 0, 1.53529571695812710730601693159, 2.42662095296569579447857421198, 3.44894263179504292792870571708, 4.41598122049440959941774048961, 5.20156982604335854050014148043, 5.45297460513828538521207937878, 6.45491054849334739547649896690, 6.94018233601734903851122908215, 8.025358327943759140158439447131

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