Properties

Label 2-4598-1.1-c1-0-121
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 − 1.61·3-s + 4-s − 1.61·5-s − 1.61·6-s + 3.85·7-s + 8-s − 0.381·9-s − 1.61·10-s − 1.61·12-s − 0.618·13-s + 3.85·14-s + 2.61·15-s + 16-s − 3.23·17-s − 0.381·18-s + 19-s − 1.61·20-s − 6.23·21-s − 2·23-s − 1.61·24-s − 2.38·25-s − 0.618·26-s + 5.47·27-s + 3.85·28-s + 0.854·29-s + 2.61·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.934·3-s + 0.5·4-s − 0.723·5-s − 0.660·6-s + 1.45·7-s + 0.353·8-s − 0.127·9-s − 0.511·10-s − 0.467·12-s − 0.171·13-s + 1.03·14-s + 0.675·15-s + 0.250·16-s − 0.784·17-s − 0.0900·18-s + 0.229·19-s − 0.361·20-s − 1.36·21-s − 0.417·23-s − 0.330·24-s − 0.476·25-s − 0.121·26-s + 1.05·27-s + 0.728·28-s + 0.158·29-s + 0.477·30-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 \( 1 + 1.61T + 3T^{2} \)
5 \( 1 + 1.61T + 5T^{2} \)
7 \( 1 - 3.85T + 7T^{2} \)
13 \( 1 + 0.618T + 13T^{2} \)
17 \( 1 + 3.23T + 17T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 - 0.854T + 29T^{2} \)
31 \( 1 + 2.61T + 31T^{2} \)
37 \( 1 - 1.23T + 37T^{2} \)
41 \( 1 - 0.618T + 41T^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 + 0.763T + 47T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 + 5.70T + 61T^{2} \)
67 \( 1 - 5.09T + 67T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 4.47T + 79T^{2} \)
83 \( 1 - 2.38T + 83T^{2} \)
89 \( 1 - 6.94T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−8.079669302764907765894245251210, −7.06641467689381830597031919905, −6.43653090598679040491518179036, −5.51707681286640136353231277711, −5.01350273666736267240843715141, −4.40756861604253388728898886618, −3.61046177234654650859244430183, −2.42482992612592506085229127537, −1.42460999369936152160007923338, 0, 1.42460999369936152160007923338, 2.42482992612592506085229127537, 3.61046177234654650859244430183, 4.40756861604253388728898886618, 5.01350273666736267240843715141, 5.51707681286640136353231277711, 6.43653090598679040491518179036, 7.06641467689381830597031919905, 8.079669302764907765894245251210

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