Properties

Label 2-98736-1.1-c1-0-49
Degree $2$
Conductor $98736$
Sign $-1$
Analytic cond. $788.410$
Root an. cond. $28.0786$
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  − 3-s − 2·5-s + 9-s + 2·13-s + 2·15-s − 17-s + 4·19-s − 25-s − 27-s + 10·29-s − 8·31-s − 2·37-s − 2·39-s − 10·41-s + 12·43-s − 2·45-s − 7·49-s + 51-s + 6·53-s − 4·57-s − 12·59-s + 10·61-s − 4·65-s + 12·67-s − 10·73-s + 75-s − 8·79-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.554·13-s + 0.516·15-s − 0.242·17-s + 0.917·19-s − 1/5·25-s − 0.192·27-s + 1.85·29-s − 1.43·31-s − 0.328·37-s − 0.320·39-s − 1.56·41-s + 1.82·43-s − 0.298·45-s − 49-s + 0.140·51-s + 0.824·53-s − 0.529·57-s − 1.56·59-s + 1.28·61-s − 0.496·65-s + 1.46·67-s − 1.17·73-s + 0.115·75-s − 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(98736\)    =    \(2^{4} \cdot 3 \cdot 11^{2} \cdot 17\)
Sign: $-1$
Analytic conductor: \(788.410\)
Root analytic conductor: \(28.0786\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 98736,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
11 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−14.00657061207413, −13.57566707504129, −12.81417860188555, −12.58474218816187, −11.90758918793706, −11.58515656895897, −11.23110202677195, −10.57307282799319, −10.25852669282105, −9.547389646840202, −9.083968624857401, −8.349900381580394, −8.136570999683867, −7.327070851354680, −7.060075290791609, −6.453281149708588, −5.794609842518784, −5.370435138971186, −4.671249372524654, −4.241672290218488, −3.551740270245009, −3.166291593306396, −2.276100325956477, −1.457185251893909, −0.7862592346574321, 0, 0.7862592346574321, 1.457185251893909, 2.276100325956477, 3.166291593306396, 3.551740270245009, 4.241672290218488, 4.671249372524654, 5.370435138971186, 5.794609842518784, 6.453281149708588, 7.060075290791609, 7.327070851354680, 8.136570999683867, 8.349900381580394, 9.083968624857401, 9.547389646840202, 10.25852669282105, 10.57307282799319, 11.23110202677195, 11.58515656895897, 11.90758918793706, 12.58474218816187, 12.81417860188555, 13.57566707504129, 14.00657061207413

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