Properties

Label 2-1045-1.1-c5-0-159
Degree $2$
Conductor $1045$
Sign $1$
Analytic cond. $167.601$
Root an. cond. $12.9460$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.779·2-s + 22.1·3-s − 31.3·4-s + 25·5-s + 17.2·6-s + 247.·7-s − 49.4·8-s + 246.·9-s + 19.4·10-s + 121·11-s − 694.·12-s − 1.10e3·13-s + 192.·14-s + 553.·15-s + 966.·16-s − 978.·17-s + 192.·18-s + 361·19-s − 784.·20-s + 5.47e3·21-s + 94.2·22-s + 552.·23-s − 1.09e3·24-s + 625·25-s − 861.·26-s + 84.4·27-s − 7.76e3·28-s + ⋯
L(s)  = 1  + 0.137·2-s + 1.41·3-s − 0.981·4-s + 0.447·5-s + 0.195·6-s + 1.90·7-s − 0.272·8-s + 1.01·9-s + 0.0616·10-s + 0.301·11-s − 1.39·12-s − 1.81·13-s + 0.262·14-s + 0.634·15-s + 0.943·16-s − 0.821·17-s + 0.139·18-s + 0.229·19-s − 0.438·20-s + 2.70·21-s + 0.0415·22-s + 0.217·23-s − 0.387·24-s + 0.200·25-s − 0.250·26-s + 0.0222·27-s − 1.87·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $1$
Analytic conductor: \(167.601\)
Root analytic conductor: \(12.9460\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(4.592418528\)
\(L(\frac12)\) \(\approx\) \(4.592418528\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - 25T \)
11 \( 1 - 121T \)
19 \( 1 - 361T \)
good2 \( 1 - 0.779T + 32T^{2} \)
3 \( 1 - 22.1T + 243T^{2} \)
7 \( 1 - 247.T + 1.68e4T^{2} \)
13 \( 1 + 1.10e3T + 3.71e5T^{2} \)
17 \( 1 + 978.T + 1.41e6T^{2} \)
23 \( 1 - 552.T + 6.43e6T^{2} \)
29 \( 1 - 6.16e3T + 2.05e7T^{2} \)
31 \( 1 - 8.83e3T + 2.86e7T^{2} \)
37 \( 1 + 1.48e4T + 6.93e7T^{2} \)
41 \( 1 + 1.53e4T + 1.15e8T^{2} \)
43 \( 1 - 1.71e4T + 1.47e8T^{2} \)
47 \( 1 - 2.14e4T + 2.29e8T^{2} \)
53 \( 1 - 2.15e4T + 4.18e8T^{2} \)
59 \( 1 - 3.29e4T + 7.14e8T^{2} \)
61 \( 1 - 4.14e4T + 8.44e8T^{2} \)
67 \( 1 + 1.42e4T + 1.35e9T^{2} \)
71 \( 1 + 3.04e4T + 1.80e9T^{2} \)
73 \( 1 + 1.99e4T + 2.07e9T^{2} \)
79 \( 1 - 4.46e4T + 3.07e9T^{2} \)
83 \( 1 - 1.13e5T + 3.93e9T^{2} \)
89 \( 1 + 4.75e4T + 5.58e9T^{2} \)
97 \( 1 + 8.69e4T + 8.58e9T^{2} \)
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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

−8.810001713743414040699042499435, −8.626393009570933697891417965930, −7.79281910211122870596342072699, −6.99606507003148397851591037002, −5.33573537317290836422272528317, −4.75952505621548301559628838300, −4.06012318709739739459959436738, −2.69776538699924998603210434280, −2.04119580414277348170814901427, −0.876523700074623690331366837659, 0.876523700074623690331366837659, 2.04119580414277348170814901427, 2.69776538699924998603210434280, 4.06012318709739739459959436738, 4.75952505621548301559628838300, 5.33573537317290836422272528317, 6.99606507003148397851591037002, 7.79281910211122870596342072699, 8.626393009570933697891417965930, 8.810001713743414040699042499435

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