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Properties

Label 2-575-1.1-c1-0-24
Degree $2$
Conductor $575$
Sign $1$
Analytic cond. $4.59139$
Root an. cond. $2.14275$
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  + 2·2-s + 2·3-s + 2·4-s + 4·6-s + 7-s + 9-s + 4·12-s + 2·13-s + 2·14-s − 4·16-s − 5·17-s + 2·18-s + 8·19-s + 2·21-s − 23-s + 4·26-s − 4·27-s + 2·28-s − 5·29-s − 5·31-s − 8·32-s − 10·34-s + 2·36-s + 7·37-s + 16·38-s + 4·39-s − 7·41-s + ⋯
L(s)  = 1  + 1.41·2-s + 1.15·3-s + 4-s + 1.63·6-s + 0.377·7-s + 1/3·9-s + 1.15·12-s + 0.554·13-s + 0.534·14-s − 16-s − 1.21·17-s + 0.471·18-s + 1.83·19-s + 0.436·21-s − 0.208·23-s + 0.784·26-s − 0.769·27-s + 0.377·28-s − 0.928·29-s − 0.898·31-s − 1.41·32-s − 1.71·34-s + 1/3·36-s + 1.15·37-s + 2.59·38-s + 0.640·39-s − 1.09·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(575\)    =    \(5^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(4.59139\)
Root analytic conductor: \(2.14275\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{575} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 575,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.064745297\)
\(L(\frac12)\) \(\approx\) \(4.064745297\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 + T \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 - 2 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 7 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 3 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−11.21645730685170637539946705503, −9.635573568718492337153683237964, −8.968247392855360442850494902852, −8.021393070095707242610426057673, −7.06279245617742916674477673238, −5.90881828705332726272674122854, −4.99590413546970297987417470463, −3.89354701982092128675905563532, −3.17036354408394678886960890665, −2.05368682964998244414756512759, 2.05368682964998244414756512759, 3.17036354408394678886960890665, 3.89354701982092128675905563532, 4.99590413546970297987417470463, 5.90881828705332726272674122854, 7.06279245617742916674477673238, 8.021393070095707242610426057673, 8.968247392855360442850494902852, 9.635573568718492337153683237964, 11.21645730685170637539946705503

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