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Properties

Label 2-80850-1.1-c1-0-9
Degree $2$
Conductor $80850$
Sign $1$
Analytic cond. $645.590$
Root an. cond. $25.4084$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 11-s + 12-s − 4·13-s + 16-s − 6·17-s − 18-s + 2·19-s − 22-s − 24-s + 4·26-s + 27-s − 6·29-s + 2·31-s − 32-s + 33-s + 6·34-s + 36-s − 2·37-s − 2·38-s − 4·39-s − 2·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.458·19-s − 0.213·22-s − 0.204·24-s + 0.784·26-s + 0.192·27-s − 1.11·29-s + 0.359·31-s − 0.176·32-s + 0.174·33-s + 1.02·34-s + 1/6·36-s − 0.328·37-s − 0.324·38-s − 0.640·39-s − 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(80850\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(645.590\)
Root analytic conductor: \(25.4084\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 80850,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.279741188\)
\(L(\frac12)\) \(\approx\) \(1.279741188\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−14.02665683879631, −13.55756256359545, −12.91377659473197, −12.52466239120389, −11.97909112610489, −11.30597343625002, −11.08654805464535, −10.27729681913625, −9.975590922588207, −9.182285668167726, −9.144067600251801, −8.557842548658477, −7.754205441212183, −7.549454626177658, −6.900994922268461, −6.515312971176308, −5.724862370274318, −5.159868955189214, −4.381389529749423, −3.986009532890914, −3.133574241389681, −2.512437765260638, −2.088163247343465, −1.333167308160445, −0.3944384104235003, 0.3944384104235003, 1.333167308160445, 2.088163247343465, 2.512437765260638, 3.133574241389681, 3.986009532890914, 4.381389529749423, 5.159868955189214, 5.724862370274318, 6.515312971176308, 6.900994922268461, 7.549454626177658, 7.754205441212183, 8.557842548658477, 9.144067600251801, 9.182285668167726, 9.975590922588207, 10.27729681913625, 11.08654805464535, 11.30597343625002, 11.97909112610489, 12.52466239120389, 12.91377659473197, 13.55756256359545, 14.02665683879631

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