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Properties

Label 2-248430-1.1-c1-0-151
Degree $2$
Conductor $248430$
Sign $-1$
Analytic cond. $1983.72$
Root an. cond. $44.5390$
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  − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s + 4·11-s − 12-s − 15-s + 16-s − 2·17-s − 18-s + 4·19-s + 20-s − 4·22-s − 8·23-s + 24-s + 25-s − 27-s − 2·29-s + 30-s − 32-s − 4·33-s + 2·34-s + 36-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.852·22-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.176·32-s − 0.696·33-s + 0.342·34-s + 1/6·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(248430\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(1983.72\)
Root analytic conductor: \(44.5390\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 248430,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 \)
13 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−12.92894689250583, −12.51665218229175, −11.96835523845333, −11.68543554719052, −11.30488694412928, −10.71772072672434, −10.24445117388130, −9.838509031094479, −9.352268086809353, −9.125428691716929, −8.337260917514946, −8.062673517648231, −7.390847097285031, −6.868946400981919, −6.413228731749327, −6.197283031911989, −5.434820207559142, −5.078795793041125, −4.408931888368453, −3.571922255617752, −3.535267107987864, −2.438115551176012, −1.873420258469074, −1.484771732673329, −0.7422826382116539, 0, 0.7422826382116539, 1.484771732673329, 1.873420258469074, 2.438115551176012, 3.535267107987864, 3.571922255617752, 4.408931888368453, 5.078795793041125, 5.434820207559142, 6.197283031911989, 6.413228731749327, 6.868946400981919, 7.390847097285031, 8.062673517648231, 8.337260917514946, 9.125428691716929, 9.352268086809353, 9.838509031094479, 10.24445117388130, 10.71772072672434, 11.30488694412928, 11.68543554719052, 11.96835523845333, 12.51665218229175, 12.92894689250583

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