Properties

Label 2-2450-1.1-c1-0-1
Degree $2$
Conductor $2450$
Sign $1$
Analytic cond. $19.5633$
Root an. cond. $4.42304$
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 − 2·3-s + 4-s + 2·6-s − 8-s + 9-s − 4·11-s − 2·12-s − 2·13-s + 16-s − 8·17-s − 18-s − 6·19-s + 4·22-s + 4·23-s + 2·24-s + 2·26-s + 4·27-s − 6·29-s + 4·31-s − 32-s + 8·33-s + 8·34-s + 36-s + 10·37-s + 6·38-s + 4·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.577·12-s − 0.554·13-s + 1/4·16-s − 1.94·17-s − 0.235·18-s − 1.37·19-s + 0.852·22-s + 0.834·23-s + 0.408·24-s + 0.392·26-s + 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 1.39·33-s + 1.37·34-s + 1/6·36-s + 1.64·37-s + 0.973·38-s + 0.640·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2450\)    =    \(2 \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(19.5633\)
Root analytic conductor: \(4.42304\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2747366797\)
\(L(\frac12)\) \(\approx\) \(0.2747366797\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 8 T + p T^{2} \)
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

−8.980008451296710170626375632277, −8.169311792616696532350927810659, −7.42579423059284725914097555874, −6.46853840054551386721568674637, −6.11996720874293404664984988869, −4.97043025034748725774602844716, −4.49802827242680952124977515011, −2.86783321468236434903187431571, −2.02264162411673129083359187598, −0.36857496855728403516977276168, 0.36857496855728403516977276168, 2.02264162411673129083359187598, 2.86783321468236434903187431571, 4.49802827242680952124977515011, 4.97043025034748725774602844716, 6.11996720874293404664984988869, 6.46853840054551386721568674637, 7.42579423059284725914097555874, 8.169311792616696532350927810659, 8.980008451296710170626375632277

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