Properties

Label 2-4598-1.1-c1-0-103
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 1.79·3-s + 4-s + 3.79·5-s − 1.79·6-s + 4.79·7-s − 8-s + 0.208·9-s − 3.79·10-s + 1.79·12-s − 1.20·13-s − 4.79·14-s + 6.79·15-s + 16-s + 7.58·17-s − 0.208·18-s − 19-s + 3.79·20-s + 8.58·21-s − 1.58·23-s − 1.79·24-s + 9.37·25-s + 1.20·26-s − 5.00·27-s + 4.79·28-s − 2.20·29-s − 6.79·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.03·3-s + 0.5·4-s + 1.69·5-s − 0.731·6-s + 1.81·7-s − 0.353·8-s + 0.0695·9-s − 1.19·10-s + 0.517·12-s − 0.335·13-s − 1.28·14-s + 1.75·15-s + 0.250·16-s + 1.83·17-s − 0.0491·18-s − 0.229·19-s + 0.847·20-s + 1.87·21-s − 0.329·23-s − 0.365·24-s + 1.87·25-s + 0.237·26-s − 0.962·27-s + 0.905·28-s − 0.410·29-s − 1.23·30-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: \(0\)
Selberg data: \((2,\ 4598,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.612932040\)
\(L(\frac12)\) \(\approx\) \(3.612932040\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.79T + 3T^{2} \)
5 \( 1 - 3.79T + 5T^{2} \)
7 \( 1 - 4.79T + 7T^{2} \)
13 \( 1 + 1.20T + 13T^{2} \)
17 \( 1 - 7.58T + 17T^{2} \)
23 \( 1 + 1.58T + 23T^{2} \)
29 \( 1 + 2.20T + 29T^{2} \)
31 \( 1 + 10.7T + 31T^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 - 0.791T + 41T^{2} \)
43 \( 1 + 7.37T + 43T^{2} \)
47 \( 1 - 9.16T + 47T^{2} \)
53 \( 1 + 1.58T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 4.79T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 9.58T + 73T^{2} \)
79 \( 1 - 5.58T + 79T^{2} \)
83 \( 1 - 3.79T + 83T^{2} \)
89 \( 1 + 4.41T + 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−8.449995309568280429974547288289, −7.64078069173246770004350448486, −7.36863679114542686209646605459, −5.87786408839645781841514074893, −5.64590425191052374461204235225, −4.73652993625233514690518052521, −3.48598730442617386988705840631, −2.48757679149046913385337091571, −1.88925969623663867404599253776, −1.28722551999193325538959718189, 1.28722551999193325538959718189, 1.88925969623663867404599253776, 2.48757679149046913385337091571, 3.48598730442617386988705840631, 4.73652993625233514690518052521, 5.64590425191052374461204235225, 5.87786408839645781841514074893, 7.36863679114542686209646605459, 7.64078069173246770004350448486, 8.449995309568280429974547288289

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