Properties

Label 2-4598-1.1-c1-0-153
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 + 2.12·3-s + 4-s − 3.12·5-s + 2.12·6-s − 0.515·7-s + 8-s + 1.51·9-s − 3.12·10-s + 2.12·12-s − 0.484·13-s − 0.515·14-s − 6.64·15-s + 16-s − 1.51·17-s + 1.51·18-s − 19-s − 3.12·20-s − 1.09·21-s + 0.515·23-s + 2.12·24-s + 4.76·25-s − 0.484·26-s − 3.15·27-s − 0.515·28-s − 2.60·29-s − 6.64·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.22·3-s + 0.5·4-s − 1.39·5-s + 0.867·6-s − 0.194·7-s + 0.353·8-s + 0.505·9-s − 0.988·10-s + 0.613·12-s − 0.134·13-s − 0.137·14-s − 1.71·15-s + 0.250·16-s − 0.367·17-s + 0.357·18-s − 0.229·19-s − 0.698·20-s − 0.238·21-s + 0.107·23-s + 0.433·24-s + 0.952·25-s − 0.0950·26-s − 0.607·27-s − 0.0973·28-s − 0.484·29-s − 1.21·30-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 \( 1 - 2.12T + 3T^{2} \)
5 \( 1 + 3.12T + 5T^{2} \)
7 \( 1 + 0.515T + 7T^{2} \)
13 \( 1 + 0.484T + 13T^{2} \)
17 \( 1 + 1.51T + 17T^{2} \)
23 \( 1 - 0.515T + 23T^{2} \)
29 \( 1 + 2.60T + 29T^{2} \)
31 \( 1 + 5.28T + 31T^{2} \)
37 \( 1 + 10.6T + 37T^{2} \)
41 \( 1 + 6.09T + 41T^{2} \)
43 \( 1 - 1.03T + 43T^{2} \)
47 \( 1 - 9.01T + 47T^{2} \)
53 \( 1 + 0.670T + 53T^{2} \)
59 \( 1 + 2.90T + 59T^{2} \)
61 \( 1 - 9.92T + 61T^{2} \)
67 \( 1 + 2.06T + 67T^{2} \)
71 \( 1 + 2.64T + 71T^{2} \)
73 \( 1 + 7.52T + 73T^{2} \)
79 \( 1 - 1.60T + 79T^{2} \)
83 \( 1 + 1.60T + 83T^{2} \)
89 \( 1 + 1.06T + 89T^{2} \)
97 \( 1 - 7.76T + 97T^{2} \)
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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

−7.88026372578262799921677515589, −7.33510801555942068517436000301, −6.74512114132157739061666052072, −5.62825749145019296587310160223, −4.75539056320370041955401718305, −3.85067331679107192728838315724, −3.55328145921969728760777201990, −2.72233440437581715290583620067, −1.77115620948359312062960589331, 0, 1.77115620948359312062960589331, 2.72233440437581715290583620067, 3.55328145921969728760777201990, 3.85067331679107192728838315724, 4.75539056320370041955401718305, 5.62825749145019296587310160223, 6.74512114132157739061666052072, 7.33510801555942068517436000301, 7.88026372578262799921677515589

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