Properties

Label 2-4598-1.1-c1-0-141
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 + 0.326·3-s + 4-s + 2.05·5-s − 0.326·6-s + 7-s − 8-s − 2.89·9-s − 2.05·10-s + 0.326·12-s + 6.11·13-s − 14-s + 0.673·15-s + 16-s − 0.239·17-s + 2.89·18-s − 19-s + 2.05·20-s + 0.326·21-s − 7.61·23-s − 0.326·24-s − 0.760·25-s − 6.11·26-s − 1.92·27-s + 28-s − 9.90·29-s − 0.673·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.188·3-s + 0.5·4-s + 0.920·5-s − 0.133·6-s + 0.377·7-s − 0.353·8-s − 0.964·9-s − 0.651·10-s + 0.0943·12-s + 1.69·13-s − 0.267·14-s + 0.173·15-s + 0.250·16-s − 0.0580·17-s + 0.681·18-s − 0.229·19-s + 0.460·20-s + 0.0713·21-s − 1.58·23-s − 0.0667·24-s − 0.152·25-s − 1.19·26-s − 0.370·27-s + 0.188·28-s − 1.84·29-s − 0.122·30-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 \( 1 - 0.326T + 3T^{2} \)
5 \( 1 - 2.05T + 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
13 \( 1 - 6.11T + 13T^{2} \)
17 \( 1 + 0.239T + 17T^{2} \)
23 \( 1 + 7.61T + 23T^{2} \)
29 \( 1 + 9.90T + 29T^{2} \)
31 \( 1 + 3.80T + 31T^{2} \)
37 \( 1 + 11.4T + 37T^{2} \)
41 \( 1 - 6.77T + 41T^{2} \)
43 \( 1 + 6.97T + 43T^{2} \)
47 \( 1 - 4.21T + 47T^{2} \)
53 \( 1 + 8.68T + 53T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 + 6.50T + 61T^{2} \)
67 \( 1 + 9.28T + 67T^{2} \)
71 \( 1 + 8.28T + 71T^{2} \)
73 \( 1 - 5.37T + 73T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 - 13.6T + 83T^{2} \)
89 \( 1 + 13.5T + 89T^{2} \)
97 \( 1 - 17.9T + 97T^{2} \)
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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

−8.130594610897071403329398947361, −7.43290586456537180330379986316, −6.33829326689684011131067476091, −5.90634494275886850279670167207, −5.35025874919038164279144063586, −3.95884234092868216880613699520, −3.27053568894918616204758428957, −2.02719851857888953669281908130, −1.62635814621842171700154043797, 0, 1.62635814621842171700154043797, 2.02719851857888953669281908130, 3.27053568894918616204758428957, 3.95884234092868216880613699520, 5.35025874919038164279144063586, 5.90634494275886850279670167207, 6.33829326689684011131067476091, 7.43290586456537180330379986316, 8.130594610897071403329398947361

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