Properties

Label 2-75-1.1-c1-0-0
Degree $2$
Conductor $75$
Sign $1$
Analytic cond. $0.598878$
Root an. cond. $0.773872$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3-s + 2·4-s − 2·6-s + 3·7-s + 9-s + 2·11-s + 2·12-s − 13-s − 6·14-s − 4·16-s − 2·17-s − 2·18-s − 5·19-s + 3·21-s − 4·22-s − 6·23-s + 2·26-s + 27-s + 6·28-s + 10·29-s − 3·31-s + 8·32-s + 2·33-s + 4·34-s + 2·36-s − 2·37-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 1.13·7-s + 1/3·9-s + 0.603·11-s + 0.577·12-s − 0.277·13-s − 1.60·14-s − 16-s − 0.485·17-s − 0.471·18-s − 1.14·19-s + 0.654·21-s − 0.852·22-s − 1.25·23-s + 0.392·26-s + 0.192·27-s + 1.13·28-s + 1.85·29-s − 0.538·31-s + 1.41·32-s + 0.348·33-s + 0.685·34-s + 1/3·36-s − 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{75} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6272349294\)
\(L(\frac12)\) \(\approx\) \(0.6272349294\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 17 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.63360099431736791767125137970, −13.71100770465530571761239213204, −12.06535678251561534347450688093, −10.88974094536361104567387978062, −9.903619965777680376182050580499, −8.655074250265874594654667643843, −8.114757177440921419069207299350, −6.77287251257420576381216593643, −4.48018036005395779311676580836, −1.90319955654708104595670502735, 1.90319955654708104595670502735, 4.48018036005395779311676580836, 6.77287251257420576381216593643, 8.114757177440921419069207299350, 8.655074250265874594654667643843, 9.903619965777680376182050580499, 10.88974094536361104567387978062, 12.06535678251561534347450688093, 13.71100770465530571761239213204, 14.63360099431736791767125137970

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