Properties

Label 2-1045-1.1-c5-0-263
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 6.89·2-s + 25.8·3-s + 15.5·4-s − 25·5-s − 178.·6-s + 221.·7-s + 113.·8-s + 423.·9-s + 172.·10-s + 121·11-s + 402.·12-s − 1.05e3·13-s − 1.52e3·14-s − 645.·15-s − 1.27e3·16-s − 1.49e3·17-s − 2.92e3·18-s + 361·19-s − 389.·20-s + 5.70e3·21-s − 834.·22-s + 4.34e3·23-s + 2.92e3·24-s + 625·25-s + 7.29e3·26-s + 4.66e3·27-s + 3.44e3·28-s + ⋯
L(s)  = 1  − 1.21·2-s + 1.65·3-s + 0.486·4-s − 0.447·5-s − 2.01·6-s + 1.70·7-s + 0.625·8-s + 1.74·9-s + 0.545·10-s + 0.301·11-s + 0.806·12-s − 1.73·13-s − 2.07·14-s − 0.740·15-s − 1.24·16-s − 1.25·17-s − 2.12·18-s + 0.229·19-s − 0.217·20-s + 2.82·21-s − 0.367·22-s + 1.71·23-s + 1.03·24-s + 0.200·25-s + 2.11·26-s + 1.23·27-s + 0.830·28-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1045,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 6.89T + 32T^{2} \)
3 \( 1 - 25.8T + 243T^{2} \)
7 \( 1 - 221.T + 1.68e4T^{2} \)
13 \( 1 + 1.05e3T + 3.71e5T^{2} \)
17 \( 1 + 1.49e3T + 1.41e6T^{2} \)
23 \( 1 - 4.34e3T + 6.43e6T^{2} \)
29 \( 1 + 4.70e3T + 2.05e7T^{2} \)
31 \( 1 + 3.14e3T + 2.86e7T^{2} \)
37 \( 1 - 1.11e4T + 6.93e7T^{2} \)
41 \( 1 + 1.35e4T + 1.15e8T^{2} \)
43 \( 1 + 1.87e3T + 1.47e8T^{2} \)
47 \( 1 + 2.82e4T + 2.29e8T^{2} \)
53 \( 1 + 1.81e4T + 4.18e8T^{2} \)
59 \( 1 + 4.48e4T + 7.14e8T^{2} \)
61 \( 1 + 2.72e4T + 8.44e8T^{2} \)
67 \( 1 + 3.10e4T + 1.35e9T^{2} \)
71 \( 1 + 3.89e4T + 1.80e9T^{2} \)
73 \( 1 + 6.49e4T + 2.07e9T^{2} \)
79 \( 1 + 1.44e4T + 3.07e9T^{2} \)
83 \( 1 - 3.81e4T + 3.93e9T^{2} \)
89 \( 1 - 7.60e4T + 5.58e9T^{2} \)
97 \( 1 + 3.70e3T + 8.58e9T^{2} \)
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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

−8.989884777984473501315615030448, −7.967972587643721026665764170202, −7.61132021924865515518258016585, −7.01533118574172348844696636422, −4.76613901591415485378838959650, −4.55674168220592260750821378023, −3.07234426362954902509230472938, −2.02126162364509271147605070410, −1.46217146989338402308000417481, 0, 1.46217146989338402308000417481, 2.02126162364509271147605070410, 3.07234426362954902509230472938, 4.55674168220592260750821378023, 4.76613901591415485378838959650, 7.01533118574172348844696636422, 7.61132021924865515518258016585, 7.967972587643721026665764170202, 8.989884777984473501315615030448

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