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Properties

Label 2-248430-1.1-c1-0-261
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 $2$

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 + 6·29-s − 30-s − 8·31-s − 32-s − 4·33-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 − 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 + 1.11·29-s − 0.182·30-s − 1.43·31-s − 0.176·32-s − 0.696·33-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: \(2\)
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 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−13.34543487371625, −13.04014036371546, −12.33009616395802, −11.96280999602699, −11.41537360138777, −10.74213460926137, −10.46855426567880, −10.01496877057143, −9.749922754126261, −8.987399620974114, −8.659661567546917, −8.282838681290569, −7.694052792708907, −7.475665533562253, −6.560410150083122, −6.468038536827033, −5.733313039151271, −5.226902043702164, −4.638215256636915, −4.029665725197424, −3.456855049432381, −2.695269600220366, −2.410750621020235, −1.802920020024841, −1.311128479279130, 0, 0, 1.311128479279130, 1.802920020024841, 2.410750621020235, 2.695269600220366, 3.456855049432381, 4.029665725197424, 4.638215256636915, 5.226902043702164, 5.733313039151271, 6.468038536827033, 6.560410150083122, 7.475665533562253, 7.694052792708907, 8.282838681290569, 8.659661567546917, 8.987399620974114, 9.749922754126261, 10.01496877057143, 10.46855426567880, 10.74213460926137, 11.41537360138777, 11.96280999602699, 12.33009616395802, 13.04014036371546, 13.34543487371625

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