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Properties

Label 2-248430-1.1-c1-0-160
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 + 2·11-s − 12-s + 15-s + 16-s + 2·17-s − 18-s − 6·19-s − 20-s − 2·22-s + 2·23-s + 24-s + 25-s − 27-s + 6·29-s − 30-s + 8·31-s − 32-s − 2·33-s − 2·34-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 + 0.603·11-s − 0.288·12-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.37·19-s − 0.223·20-s − 0.426·22-s + 0.417·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.182·30-s + 1.43·31-s − 0.176·32-s − 0.348·33-s − 0.342·34-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 8 T + p T^{2} \)
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

−12.93745752613011, −12.44170136429360, −11.95048306510344, −11.80391623435097, −11.25123939378080, −10.65224251937377, −10.32635459933722, −10.03287658014843, −9.300594673577436, −8.909444613566658, −8.323186313755576, −8.090859485817391, −7.472479885553710, −6.818725797360712, −6.504968501970709, −6.225064623626455, −5.477880036852838, −4.690800915234416, −4.648075088212527, −3.705949274087753, −3.358923167844875, −2.566265412609481, −1.997228710823957, −1.232547310668597, −0.7579910991029636, 0, 0.7579910991029636, 1.232547310668597, 1.997228710823957, 2.566265412609481, 3.358923167844875, 3.705949274087753, 4.648075088212527, 4.690800915234416, 5.477880036852838, 6.225064623626455, 6.504968501970709, 6.818725797360712, 7.472479885553710, 8.090859485817391, 8.323186313755576, 8.909444613566658, 9.300594673577436, 10.03287658014843, 10.32635459933722, 10.65224251937377, 11.25123939378080, 11.80391623435097, 11.95048306510344, 12.44170136429360, 12.93745752613011

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