Properties

Label 2-2450-1.1-c1-0-7
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 + 4-s − 8-s − 3·9-s + 4·11-s − 6·13-s + 16-s + 2·17-s + 3·18-s − 4·22-s + 6·26-s + 6·29-s − 8·31-s − 32-s − 2·34-s − 3·36-s + 10·37-s − 2·41-s − 4·43-s + 4·44-s + 8·47-s − 6·52-s + 2·53-s − 6·58-s + 8·59-s + 14·61-s + 8·62-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s + 1.20·11-s − 1.66·13-s + 1/4·16-s + 0.485·17-s + 0.707·18-s − 0.852·22-s + 1.17·26-s + 1.11·29-s − 1.43·31-s − 0.176·32-s − 0.342·34-s − 1/2·36-s + 1.64·37-s − 0.312·41-s − 0.609·43-s + 0.603·44-s + 1.16·47-s − 0.832·52-s + 0.274·53-s − 0.787·58-s + 1.04·59-s + 1.79·61-s + 1.01·62-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

\(L(1)\) \(\approx\) \(1.059736022\)
\(L(\frac12)\) \(\approx\) \(1.059736022\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−8.947791175446811247051314576506, −8.301913068586907274040785472708, −7.42472757379966149042918376603, −6.82123838368350264589622397869, −5.91548647651224347146838875772, −5.14037714155681614978134696209, −4.03694373686491315845893969603, −2.95779572995708486306206498227, −2.11286590149397096544933120998, −0.71479745199007503910946566518, 0.71479745199007503910946566518, 2.11286590149397096544933120998, 2.95779572995708486306206498227, 4.03694373686491315845893969603, 5.14037714155681614978134696209, 5.91548647651224347146838875772, 6.82123838368350264589622397869, 7.42472757379966149042918376603, 8.301913068586907274040785472708, 8.947791175446811247051314576506

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