Properties

Label 2-4598-1.1-c1-0-154
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 + 2.23·3-s + 4-s − 1.61·5-s + 2.23·6-s − 3.85·7-s + 8-s + 2.00·9-s − 1.61·10-s + 2.23·12-s − 0.618·13-s − 3.85·14-s − 3.61·15-s + 16-s − 1.76·17-s + 2.00·18-s + 19-s − 1.61·20-s − 8.61·21-s + 4.23·23-s + 2.23·24-s − 2.38·25-s − 0.618·26-s − 2.23·27-s − 3.85·28-s − 1.52·29-s − 3.61·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.29·3-s + 0.5·4-s − 0.723·5-s + 0.912·6-s − 1.45·7-s + 0.353·8-s + 0.666·9-s − 0.511·10-s + 0.645·12-s − 0.171·13-s − 1.03·14-s − 0.934·15-s + 0.250·16-s − 0.427·17-s + 0.471·18-s + 0.229·19-s − 0.361·20-s − 1.88·21-s + 0.883·23-s + 0.456·24-s − 0.476·25-s − 0.121·26-s − 0.430·27-s − 0.728·28-s − 0.283·29-s − 0.660·30-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 \( 1 - 2.23T + 3T^{2} \)
5 \( 1 + 1.61T + 5T^{2} \)
7 \( 1 + 3.85T + 7T^{2} \)
13 \( 1 + 0.618T + 13T^{2} \)
17 \( 1 + 1.76T + 17T^{2} \)
23 \( 1 - 4.23T + 23T^{2} \)
29 \( 1 + 1.52T + 29T^{2} \)
31 \( 1 - 2.70T + 31T^{2} \)
37 \( 1 + 1.14T + 37T^{2} \)
41 \( 1 + 7.09T + 41T^{2} \)
43 \( 1 + 9.70T + 43T^{2} \)
47 \( 1 - 2.52T + 47T^{2} \)
53 \( 1 + 11.9T + 53T^{2} \)
59 \( 1 + 1.90T + 59T^{2} \)
61 \( 1 + 6.61T + 61T^{2} \)
67 \( 1 - 5.09T + 67T^{2} \)
71 \( 1 + 8.23T + 71T^{2} \)
73 \( 1 + 14.3T + 73T^{2} \)
79 \( 1 - 3T + 79T^{2} \)
83 \( 1 + 10.0T + 83T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
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

−7.938311287615067027256450119284, −7.19067486828980283723250443888, −6.64884586895535107843347327122, −5.80450665565625033131947837239, −4.74843159085696755802413995431, −3.93915775758761204684414141910, −3.16863260806119009757599960044, −2.97583869090126255660477465309, −1.77857402604288117919910419113, 0, 1.77857402604288117919910419113, 2.97583869090126255660477465309, 3.16863260806119009757599960044, 3.93915775758761204684414141910, 4.74843159085696755802413995431, 5.80450665565625033131947837239, 6.64884586895535107843347327122, 7.19067486828980283723250443888, 7.938311287615067027256450119284

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