Properties

Label 2-4598-1.1-c1-0-133
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 + 2·5-s − 6-s − 3·7-s + 8-s − 2·9-s + 2·10-s − 12-s + 2·13-s − 3·14-s − 2·15-s + 16-s + 2·17-s − 2·18-s − 19-s + 2·20-s + 3·21-s − 23-s − 24-s − 25-s + 2·26-s + 5·27-s − 3·28-s − 5·29-s − 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s − 2/3·9-s + 0.632·10-s − 0.288·12-s + 0.554·13-s − 0.801·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.471·18-s − 0.229·19-s + 0.447·20-s + 0.654·21-s − 0.208·23-s − 0.204·24-s − 1/5·25-s + 0.392·26-s + 0.962·27-s − 0.566·28-s − 0.928·29-s − 0.365·30-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: 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 \( 1 - 2 T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−7.78151218288077478977168753185, −6.93584518468912270900995746603, −6.15364013153051301443253467181, −5.84051037869200021796884427587, −5.30812815840586472937784833666, −4.16020356360102787822054571441, −3.35469611081304078184838017364, −2.59902861333584776243201559533, −1.52581543186035418373331338991, 0, 1.52581543186035418373331338991, 2.59902861333584776243201559533, 3.35469611081304078184838017364, 4.16020356360102787822054571441, 5.30812815840586472937784833666, 5.84051037869200021796884427587, 6.15364013153051301443253467181, 6.93584518468912270900995746603, 7.78151218288077478977168753185

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