Properties

Label 2-4598-1.1-c1-0-102
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 + 2.79·3-s + 4-s + 4.02·5-s − 2.79·6-s − 0.274·7-s − 8-s + 4.79·9-s − 4.02·10-s + 2.79·12-s − 0.597·13-s + 0.274·14-s + 11.2·15-s + 16-s + 2.62·17-s − 4.79·18-s + 19-s + 4.02·20-s − 0.767·21-s + 4.04·23-s − 2.79·24-s + 11.1·25-s + 0.597·26-s + 4.99·27-s − 0.274·28-s + 7.29·29-s − 11.2·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.61·3-s + 0.5·4-s + 1.79·5-s − 1.13·6-s − 0.103·7-s − 0.353·8-s + 1.59·9-s − 1.27·10-s + 0.805·12-s − 0.165·13-s + 0.0734·14-s + 2.89·15-s + 0.250·16-s + 0.637·17-s − 1.12·18-s + 0.229·19-s + 0.899·20-s − 0.167·21-s + 0.843·23-s − 0.569·24-s + 2.23·25-s + 0.117·26-s + 0.962·27-s − 0.0519·28-s + 1.35·29-s − 2.05·30-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

\(L(1)\) \(\approx\) \(3.887088301\)
\(L(\frac12)\) \(\approx\) \(3.887088301\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.79T + 3T^{2} \)
5 \( 1 - 4.02T + 5T^{2} \)
7 \( 1 + 0.274T + 7T^{2} \)
13 \( 1 + 0.597T + 13T^{2} \)
17 \( 1 - 2.62T + 17T^{2} \)
23 \( 1 - 4.04T + 23T^{2} \)
29 \( 1 - 7.29T + 29T^{2} \)
31 \( 1 + 1.48T + 31T^{2} \)
37 \( 1 + 8.31T + 37T^{2} \)
41 \( 1 - 12.3T + 41T^{2} \)
43 \( 1 + 5.58T + 43T^{2} \)
47 \( 1 + 9.10T + 47T^{2} \)
53 \( 1 - 7.30T + 53T^{2} \)
59 \( 1 + 7.53T + 59T^{2} \)
61 \( 1 + 4.83T + 61T^{2} \)
67 \( 1 + 14.0T + 67T^{2} \)
71 \( 1 + 9.61T + 71T^{2} \)
73 \( 1 - 3.15T + 73T^{2} \)
79 \( 1 - 2.42T + 79T^{2} \)
83 \( 1 + 15.7T + 83T^{2} \)
89 \( 1 - 1.49T + 89T^{2} \)
97 \( 1 - 9.00T + 97T^{2} \)
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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

−8.542059510862506552252505549477, −7.76643546484801317608759687461, −7.03375446565863422671825031967, −6.33602629270980713671825421091, −5.50160891405612213977670471675, −4.59355905372068623433634471182, −3.16725376968171598884815843166, −2.85062265538887166126993282975, −1.89875889866990554619032283438, −1.28502238627651904527915799619, 1.28502238627651904527915799619, 1.89875889866990554619032283438, 2.85062265538887166126993282975, 3.16725376968171598884815843166, 4.59355905372068623433634471182, 5.50160891405612213977670471675, 6.33602629270980713671825421091, 7.03375446565863422671825031967, 7.76643546484801317608759687461, 8.542059510862506552252505549477

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