Properties

Label 2-1045-1.1-c5-0-106
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  + 4.88·2-s − 26.8·3-s − 8.14·4-s − 25·5-s − 130.·6-s − 171.·7-s − 196.·8-s + 475.·9-s − 122.·10-s − 121·11-s + 218.·12-s − 567.·13-s − 836.·14-s + 670.·15-s − 696.·16-s − 666.·17-s + 2.32e3·18-s − 361·19-s + 203.·20-s + 4.58e3·21-s − 590.·22-s + 2.08e3·23-s + 5.25e3·24-s + 625·25-s − 2.77e3·26-s − 6.23e3·27-s + 1.39e3·28-s + ⋯
L(s)  = 1  + 0.863·2-s − 1.71·3-s − 0.254·4-s − 0.447·5-s − 1.48·6-s − 1.32·7-s − 1.08·8-s + 1.95·9-s − 0.386·10-s − 0.301·11-s + 0.437·12-s − 0.932·13-s − 1.14·14-s + 0.769·15-s − 0.680·16-s − 0.559·17-s + 1.68·18-s − 0.229·19-s + 0.113·20-s + 2.27·21-s − 0.260·22-s + 0.823·23-s + 1.86·24-s + 0.200·25-s − 0.804·26-s − 1.64·27-s + 0.336·28-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 \( 1 - 4.88T + 32T^{2} \)
3 \( 1 + 26.8T + 243T^{2} \)
7 \( 1 + 171.T + 1.68e4T^{2} \)
13 \( 1 + 567.T + 3.71e5T^{2} \)
17 \( 1 + 666.T + 1.41e6T^{2} \)
23 \( 1 - 2.08e3T + 6.43e6T^{2} \)
29 \( 1 + 497.T + 2.05e7T^{2} \)
31 \( 1 + 6.08e3T + 2.86e7T^{2} \)
37 \( 1 - 1.81e3T + 6.93e7T^{2} \)
41 \( 1 - 1.16e4T + 1.15e8T^{2} \)
43 \( 1 - 1.19e4T + 1.47e8T^{2} \)
47 \( 1 - 2.78e4T + 2.29e8T^{2} \)
53 \( 1 + 3.24e4T + 4.18e8T^{2} \)
59 \( 1 + 2.37e4T + 7.14e8T^{2} \)
61 \( 1 - 1.20e4T + 8.44e8T^{2} \)
67 \( 1 - 3.92e4T + 1.35e9T^{2} \)
71 \( 1 - 2.46e4T + 1.80e9T^{2} \)
73 \( 1 - 3.45e4T + 2.07e9T^{2} \)
79 \( 1 + 9.20e4T + 3.07e9T^{2} \)
83 \( 1 - 1.01e5T + 3.93e9T^{2} \)
89 \( 1 + 2.05e4T + 5.58e9T^{2} \)
97 \( 1 - 1.65e4T + 8.58e9T^{2} \)
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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

−9.094417744719522498712343568805, −7.51683221644016794897875866388, −6.73033631909300414897948958990, −6.03832757103682701991960339544, −5.31738503205391938164131131622, −4.56818086616667259321375448572, −3.77581363518279254461849944913, −2.62619894125710868787425886643, −0.65558201788713057525970769549, 0, 0.65558201788713057525970769549, 2.62619894125710868787425886643, 3.77581363518279254461849944913, 4.56818086616667259321375448572, 5.31738503205391938164131131622, 6.03832757103682701991960339544, 6.73033631909300414897948958990, 7.51683221644016794897875866388, 9.094417744719522498712343568805

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