Properties

Label 2-75-1.1-c7-0-14
Degree $2$
Conductor $75$
Sign $-1$
Analytic cond. $23.4288$
Root an. cond. $4.84033$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 15.1·2-s + 27·3-s + 101.·4-s − 408.·6-s − 198.·7-s + 407.·8-s + 729·9-s − 5.26e3·11-s + 2.72e3·12-s − 1.21e3·13-s + 3.01e3·14-s − 1.91e4·16-s + 3.45e4·17-s − 1.10e4·18-s + 1.86e4·19-s − 5.37e3·21-s + 7.97e4·22-s + 3.33e4·23-s + 1.09e4·24-s + 1.83e4·26-s + 1.96e4·27-s − 2.01e4·28-s − 1.78e5·29-s − 2.37e5·31-s + 2.37e5·32-s − 1.42e5·33-s − 5.23e5·34-s + ⋯
L(s)  = 1  − 1.33·2-s + 0.577·3-s + 0.789·4-s − 0.772·6-s − 0.219·7-s + 0.281·8-s + 0.333·9-s − 1.19·11-s + 0.455·12-s − 0.153·13-s + 0.293·14-s − 1.16·16-s + 1.70·17-s − 0.445·18-s + 0.622·19-s − 0.126·21-s + 1.59·22-s + 0.571·23-s + 0.162·24-s + 0.204·26-s + 0.192·27-s − 0.173·28-s − 1.35·29-s − 1.43·31-s + 1.27·32-s − 0.688·33-s − 2.28·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(23.4288\)
Root analytic conductor: \(4.84033\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 75,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 27T \)
5 \( 1 \)
good2 \( 1 + 15.1T + 128T^{2} \)
7 \( 1 + 198.T + 8.23e5T^{2} \)
11 \( 1 + 5.26e3T + 1.94e7T^{2} \)
13 \( 1 + 1.21e3T + 6.27e7T^{2} \)
17 \( 1 - 3.45e4T + 4.10e8T^{2} \)
19 \( 1 - 1.86e4T + 8.93e8T^{2} \)
23 \( 1 - 3.33e4T + 3.40e9T^{2} \)
29 \( 1 + 1.78e5T + 1.72e10T^{2} \)
31 \( 1 + 2.37e5T + 2.75e10T^{2} \)
37 \( 1 + 4.82e5T + 9.49e10T^{2} \)
41 \( 1 - 2.93e5T + 1.94e11T^{2} \)
43 \( 1 + 4.43e5T + 2.71e11T^{2} \)
47 \( 1 - 4.81e4T + 5.06e11T^{2} \)
53 \( 1 + 1.66e6T + 1.17e12T^{2} \)
59 \( 1 - 1.75e6T + 2.48e12T^{2} \)
61 \( 1 + 3.15e6T + 3.14e12T^{2} \)
67 \( 1 + 2.29e6T + 6.06e12T^{2} \)
71 \( 1 + 2.71e6T + 9.09e12T^{2} \)
73 \( 1 - 2.67e6T + 1.10e13T^{2} \)
79 \( 1 - 3.44e6T + 1.92e13T^{2} \)
83 \( 1 + 1.71e6T + 2.71e13T^{2} \)
89 \( 1 - 3.52e6T + 4.42e13T^{2} \)
97 \( 1 + 1.44e7T + 8.07e13T^{2} \)
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

−12.55555259579185137154124662171, −10.96815576616069821088150632545, −9.995768414766857779337021562429, −9.190932805426060003574524924154, −7.952878563514913985262948339296, −7.31565737839715962067225901613, −5.28527509469159864412530517113, −3.23518063942483111155122389749, −1.60175371726064610287877131697, 0, 1.60175371726064610287877131697, 3.23518063942483111155122389749, 5.28527509469159864412530517113, 7.31565737839715962067225901613, 7.952878563514913985262948339296, 9.190932805426060003574524924154, 9.995768414766857779337021562429, 10.96815576616069821088150632545, 12.55555259579185137154124662171

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