Properties

Label 2-4598-1.1-c1-0-120
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 + 0.158·3-s + 4-s + 3.07·5-s − 0.158·6-s − 1.35·7-s − 8-s − 2.97·9-s − 3.07·10-s + 0.158·12-s + 0.526·13-s + 1.35·14-s + 0.488·15-s + 16-s − 7.23·17-s + 2.97·18-s + 19-s + 3.07·20-s − 0.214·21-s + 2.09·23-s − 0.158·24-s + 4.46·25-s − 0.526·26-s − 0.948·27-s − 1.35·28-s + 0.277·29-s − 0.488·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.0916·3-s + 0.5·4-s + 1.37·5-s − 0.0648·6-s − 0.511·7-s − 0.353·8-s − 0.991·9-s − 0.973·10-s + 0.0458·12-s + 0.146·13-s + 0.361·14-s + 0.126·15-s + 0.250·16-s − 1.75·17-s + 0.701·18-s + 0.229·19-s + 0.688·20-s − 0.0468·21-s + 0.436·23-s − 0.0324·24-s + 0.893·25-s − 0.103·26-s − 0.182·27-s − 0.255·28-s + 0.0515·29-s − 0.0891·30-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: Trivial
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 \( 1 - 0.158T + 3T^{2} \)
5 \( 1 - 3.07T + 5T^{2} \)
7 \( 1 + 1.35T + 7T^{2} \)
13 \( 1 - 0.526T + 13T^{2} \)
17 \( 1 + 7.23T + 17T^{2} \)
23 \( 1 - 2.09T + 23T^{2} \)
29 \( 1 - 0.277T + 29T^{2} \)
31 \( 1 + 1.72T + 31T^{2} \)
37 \( 1 - 8.91T + 37T^{2} \)
41 \( 1 - 8.37T + 41T^{2} \)
43 \( 1 + 6.30T + 43T^{2} \)
47 \( 1 - 1.96T + 47T^{2} \)
53 \( 1 + 2.20T + 53T^{2} \)
59 \( 1 - 8.72T + 59T^{2} \)
61 \( 1 + 6.38T + 61T^{2} \)
67 \( 1 + 1.02T + 67T^{2} \)
71 \( 1 + 3.71T + 71T^{2} \)
73 \( 1 + 1.72T + 73T^{2} \)
79 \( 1 + 7.53T + 79T^{2} \)
83 \( 1 + 6.49T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 10.4T + 97T^{2} \)
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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

−8.210709373708600574632292528055, −7.14768363732364721155122665426, −6.45130762980135111522236063296, −5.97410453150298184665247302060, −5.26353323774928263631667093809, −4.16872069938312179979479454655, −2.85771938702019310486390732629, −2.45977213814147081954855718044, −1.41630660413439487458459463387, 0, 1.41630660413439487458459463387, 2.45977213814147081954855718044, 2.85771938702019310486390732629, 4.16872069938312179979479454655, 5.26353323774928263631667093809, 5.97410453150298184665247302060, 6.45130762980135111522236063296, 7.14768363732364721155122665426, 8.210709373708600574632292528055

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