Properties

Label 2-4598-1.1-c1-0-116
Degree $2$
Conductor $4598$
Sign $-1$
Analytic cond. $36.7152$
Root an. cond. $6.05930$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s − 2·9-s + 12-s + 2·13-s + 14-s + 16-s + 6·17-s + 2·18-s + 19-s − 21-s − 9·23-s − 24-s − 5·25-s − 2·26-s − 5·27-s − 28-s + 3·29-s + 8·31-s − 32-s − 6·34-s − 2·36-s − 7·37-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s + 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.471·18-s + 0.229·19-s − 0.218·21-s − 1.87·23-s − 0.204·24-s − 25-s − 0.392·26-s − 0.962·27-s − 0.188·28-s + 0.557·29-s + 1.43·31-s − 0.176·32-s − 1.02·34-s − 1/3·36-s − 1.15·37-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: yes
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 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 9 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 7 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 9 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−8.184206734061753315129665023257, −7.56119198074043944042548784343, −6.45551106707916369319201081980, −5.99964447663547060997519699177, −5.15064908967874935587499015081, −3.81940579301223498865955736204, −3.28657802815929334491983585503, −2.38132054649060130995470312921, −1.38173049019092216490553795461, 0, 1.38173049019092216490553795461, 2.38132054649060130995470312921, 3.28657802815929334491983585503, 3.81940579301223498865955736204, 5.15064908967874935587499015081, 5.99964447663547060997519699177, 6.45551106707916369319201081980, 7.56119198074043944042548784343, 8.184206734061753315129665023257

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