Properties

Degree $2$
Conductor $2450$
Sign $1$
Motivic weight $1$
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: $1$
Motivic weight: \(1\)
Character: $\chi_{2450} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
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

−19.16243112769306, −18.32839667558208, −17.57741836702326, −17.11576560830426, −16.69395894474263, −16.05011064858413, −14.94567285237701, −14.58115265030327, −14.15554036852377, −13.02312990151244, −12.21015206997766, −11.73226621115577, −11.17421154634321, −10.16423138714631, −9.651440390972516, −8.947791175446811, −8.301913068586907, −7.424727573799661, −6.821238383683503, −5.915486476512243, −5.140377141556816, −4.036943736864913, −2.957795729957085, −2.112865901493971, −0.7147974519900750, 0.7147974519900750, 2.112865901493971, 2.957795729957085, 4.036943736864913, 5.140377141556816, 5.915486476512243, 6.821238383683503, 7.424727573799661, 8.301913068586907, 8.947791175446811, 9.651440390972516, 10.16423138714631, 11.17421154634321, 11.73226621115577, 12.21015206997766, 13.02312990151244, 14.15554036852377, 14.58115265030327, 14.94567285237701, 16.05011064858413, 16.69395894474263, 17.11576560830426, 17.57741836702326, 18.32839667558208, 19.16243112769306

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