Properties

Label 2-2450-1.1-c1-0-19
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 − 3-s + 4-s − 6-s + 8-s − 2·9-s + 3·11-s − 12-s − 2·13-s + 16-s − 3·17-s − 2·18-s + 7·19-s + 3·22-s − 24-s − 2·26-s + 5·27-s − 6·29-s + 4·31-s + 32-s − 3·33-s − 3·34-s − 2·36-s + 8·37-s + 7·38-s + 2·39-s + 9·41-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 − 0.554·13-s + 1/4·16-s − 0.727·17-s − 0.471·18-s + 1.60·19-s + 0.639·22-s − 0.204·24-s − 0.392·26-s + 0.962·27-s − 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.522·33-s − 0.514·34-s − 1/3·36-s + 1.31·37-s + 1.13·38-s + 0.320·39-s + 1.40·41-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: $\chi_{2450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.255138778\)
\(L(\frac12)\) \(\approx\) \(2.255138778\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 5 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
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

−9.221701692759308985116864892196, −7.924849956818651360533058343824, −7.31618067193530795960620665696, −6.33657516384953168545460383012, −5.86978936708835719235551167417, −5.00240257185710799704212008518, −4.26777098613754301708772688504, −3.25417834090155218803572132829, −2.33477820265690219389580463783, −0.899513444547971617919916680990, 0.899513444547971617919916680990, 2.33477820265690219389580463783, 3.25417834090155218803572132829, 4.26777098613754301708772688504, 5.00240257185710799704212008518, 5.86978936708835719235551167417, 6.33657516384953168545460383012, 7.31618067193530795960620665696, 7.924849956818651360533058343824, 9.221701692759308985116864892196

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