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Properties

Label 2-28498-1.1-c1-0-2
Degree $2$
Conductor $28498$
Sign $-1$
Analytic cond. $227.557$
Root an. cond. $15.0850$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $3$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s + 4-s − 2·5-s + 3·6-s − 2·7-s − 8-s + 6·9-s + 2·10-s − 6·11-s − 3·12-s − 5·13-s + 2·14-s + 6·15-s + 16-s − 6·17-s − 6·18-s − 5·19-s − 2·20-s + 6·21-s + 6·22-s − 4·23-s + 3·24-s − 25-s + 5·26-s − 9·27-s − 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.894·5-s + 1.22·6-s − 0.755·7-s − 0.353·8-s + 2·9-s + 0.632·10-s − 1.80·11-s − 0.866·12-s − 1.38·13-s + 0.534·14-s + 1.54·15-s + 1/4·16-s − 1.45·17-s − 1.41·18-s − 1.14·19-s − 0.447·20-s + 1.30·21-s + 1.27·22-s − 0.834·23-s + 0.612·24-s − 1/5·25-s + 0.980·26-s − 1.73·27-s − 0.377·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28498 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28498 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28498\)    =    \(2 \cdot 14249\)
Sign: $-1$
Analytic conductor: \(227.557\)
Root analytic conductor: \(15.0850\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((2,\ 28498,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
14249 \( 1 + T \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 5 T + p T^{2} \)
83 \( 1 + 5 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
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

−15.98037465792873, −15.71671756544404, −15.27549162990950, −14.82001793347109, −13.53859796712482, −13.13728780804175, −12.53763921928773, −12.21270857643398, −11.64251891653310, −11.13773715234147, −10.52657991076812, −10.31060228706013, −9.788223142791099, −8.945373926283372, −8.277519250155768, −7.599492877548185, −7.132222945744085, −6.716907248767030, −6.006188793229957, −5.399118657977578, −4.836992614403974, −4.254177177597457, −3.399732336611203, −2.387501682302575, −1.815001863130046, 0, 0, 0, 1.815001863130046, 2.387501682302575, 3.399732336611203, 4.254177177597457, 4.836992614403974, 5.399118657977578, 6.006188793229957, 6.716907248767030, 7.132222945744085, 7.599492877548185, 8.277519250155768, 8.945373926283372, 9.788223142791099, 10.31060228706013, 10.52657991076812, 11.13773715234147, 11.64251891653310, 12.21270857643398, 12.53763921928773, 13.13728780804175, 13.53859796712482, 14.82001793347109, 15.27549162990950, 15.71671756544404, 15.98037465792873

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