Properties

Label 2-1075-1.1-c5-0-13
Degree $2$
Conductor $1075$
Sign $1$
Analytic cond. $172.412$
Root an. cond. $13.1305$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 8.09·2-s − 11.1·3-s + 33.5·4-s − 90.5·6-s − 223.·7-s + 12.3·8-s − 118.·9-s − 631.·11-s − 374.·12-s − 28.5·13-s − 1.80e3·14-s − 972.·16-s + 1.74e3·17-s − 955.·18-s − 2.02e3·19-s + 2.49e3·21-s − 5.11e3·22-s − 2.98e3·23-s − 138.·24-s − 231.·26-s + 4.03e3·27-s − 7.49e3·28-s + 766.·29-s − 8.35e3·31-s − 8.27e3·32-s + 7.06e3·33-s + 1.41e4·34-s + ⋯
L(s)  = 1  + 1.43·2-s − 0.717·3-s + 1.04·4-s − 1.02·6-s − 1.72·7-s + 0.0684·8-s − 0.485·9-s − 1.57·11-s − 0.751·12-s − 0.0468·13-s − 2.46·14-s − 0.949·16-s + 1.46·17-s − 0.694·18-s − 1.28·19-s + 1.23·21-s − 2.25·22-s − 1.17·23-s − 0.0490·24-s − 0.0671·26-s + 1.06·27-s − 1.80·28-s + 0.169·29-s − 1.56·31-s − 1.42·32-s + 1.12·33-s + 2.09·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1075\)    =    \(5^{2} \cdot 43\)
Sign: $1$
Analytic conductor: \(172.412\)
Root analytic conductor: \(13.1305\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{1075} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1075,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.1495530015\)
\(L(\frac12)\) \(\approx\) \(0.1495530015\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
43 \( 1 - 1.84e3T \)
good2 \( 1 - 8.09T + 32T^{2} \)
3 \( 1 + 11.1T + 243T^{2} \)
7 \( 1 + 223.T + 1.68e4T^{2} \)
11 \( 1 + 631.T + 1.61e5T^{2} \)
13 \( 1 + 28.5T + 3.71e5T^{2} \)
17 \( 1 - 1.74e3T + 1.41e6T^{2} \)
19 \( 1 + 2.02e3T + 2.47e6T^{2} \)
23 \( 1 + 2.98e3T + 6.43e6T^{2} \)
29 \( 1 - 766.T + 2.05e7T^{2} \)
31 \( 1 + 8.35e3T + 2.86e7T^{2} \)
37 \( 1 + 1.48e4T + 6.93e7T^{2} \)
41 \( 1 + 5.34e3T + 1.15e8T^{2} \)
47 \( 1 - 6.28e3T + 2.29e8T^{2} \)
53 \( 1 - 915.T + 4.18e8T^{2} \)
59 \( 1 + 1.46e4T + 7.14e8T^{2} \)
61 \( 1 + 2.13e4T + 8.44e8T^{2} \)
67 \( 1 - 1.28e4T + 1.35e9T^{2} \)
71 \( 1 - 5.64e4T + 1.80e9T^{2} \)
73 \( 1 - 2.55e4T + 2.07e9T^{2} \)
79 \( 1 - 5.79e3T + 3.07e9T^{2} \)
83 \( 1 - 7.85e3T + 3.93e9T^{2} \)
89 \( 1 + 7.56e3T + 5.58e9T^{2} \)
97 \( 1 + 1.11e5T + 8.58e9T^{2} \)
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

−9.302293147776316925909031362628, −8.195171977699894605391682014747, −7.04174857165900529977599627608, −6.23231034433175213282700791143, −5.64049096646379812326626869299, −5.12370441034214503017804453732, −3.80782761194455085080324955257, −3.17335276863640677495860993190, −2.29993832711636895252314353432, −0.12520119408309380952148752734, 0.12520119408309380952148752734, 2.29993832711636895252314353432, 3.17335276863640677495860993190, 3.80782761194455085080324955257, 5.12370441034214503017804453732, 5.64049096646379812326626869299, 6.23231034433175213282700791143, 7.04174857165900529977599627608, 8.195171977699894605391682014747, 9.302293147776316925909031362628

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