Properties

Label 2-2450-1.1-c1-0-63
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 6-s + 8-s − 2·9-s − 3·11-s + 12-s − 4·13-s + 16-s − 3·17-s − 2·18-s − 5·19-s − 3·22-s − 6·23-s + 24-s − 4·26-s − 5·27-s − 2·31-s + 32-s − 3·33-s − 3·34-s − 2·36-s − 2·37-s − 5·38-s − 4·39-s + 3·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.904·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 1.14·19-s − 0.639·22-s − 1.25·23-s + 0.204·24-s − 0.784·26-s − 0.962·27-s − 0.359·31-s + 0.176·32-s − 0.522·33-s − 0.514·34-s − 1/3·36-s − 0.328·37-s − 0.811·38-s − 0.640·39-s + 0.468·41-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: \(1\)
Selberg data: \((2,\ 2450,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 - T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
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

−8.409810000821400938912682522137, −7.82315988608083832795544283712, −7.03565158477587046360278529923, −6.08705351503109341254138790112, −5.38603857261488641210316347351, −4.49973313906049110134055340472, −3.70222391402935277847702964230, −2.50196990522414289793307495614, −2.23496341365700158267997210371, 0, 2.23496341365700158267997210371, 2.50196990522414289793307495614, 3.70222391402935277847702964230, 4.49973313906049110134055340472, 5.38603857261488641210316347351, 6.08705351503109341254138790112, 7.03565158477587046360278529923, 7.82315988608083832795544283712, 8.409810000821400938912682522137

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