Properties

Label 2-1045-1.1-c5-0-287
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  + 9.38·2-s + 13.7·3-s + 56.0·4-s − 25·5-s + 129.·6-s − 129.·7-s + 225.·8-s − 53.7·9-s − 234.·10-s + 121·11-s + 770.·12-s + 848.·13-s − 1.21e3·14-s − 343.·15-s + 320.·16-s − 1.28e3·17-s − 504.·18-s + 361·19-s − 1.40e3·20-s − 1.78e3·21-s + 1.13e3·22-s + 741.·23-s + 3.09e3·24-s + 625·25-s + 7.96e3·26-s − 4.08e3·27-s − 7.25e3·28-s + ⋯
L(s)  = 1  + 1.65·2-s + 0.882·3-s + 1.75·4-s − 0.447·5-s + 1.46·6-s − 0.999·7-s + 1.24·8-s − 0.221·9-s − 0.741·10-s + 0.301·11-s + 1.54·12-s + 1.39·13-s − 1.65·14-s − 0.394·15-s + 0.313·16-s − 1.07·17-s − 0.366·18-s + 0.229·19-s − 0.782·20-s − 0.881·21-s + 0.500·22-s + 0.292·23-s + 1.09·24-s + 0.200·25-s + 2.31·26-s − 1.07·27-s − 1.74·28-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 \( 1 - 9.38T + 32T^{2} \)
3 \( 1 - 13.7T + 243T^{2} \)
7 \( 1 + 129.T + 1.68e4T^{2} \)
13 \( 1 - 848.T + 3.71e5T^{2} \)
17 \( 1 + 1.28e3T + 1.41e6T^{2} \)
23 \( 1 - 741.T + 6.43e6T^{2} \)
29 \( 1 + 5.03e3T + 2.05e7T^{2} \)
31 \( 1 + 2.87e3T + 2.86e7T^{2} \)
37 \( 1 + 5.40e3T + 6.93e7T^{2} \)
41 \( 1 + 1.06e4T + 1.15e8T^{2} \)
43 \( 1 - 1.54e4T + 1.47e8T^{2} \)
47 \( 1 + 1.42e4T + 2.29e8T^{2} \)
53 \( 1 + 1.38e4T + 4.18e8T^{2} \)
59 \( 1 + 1.77e4T + 7.14e8T^{2} \)
61 \( 1 - 1.02e4T + 8.44e8T^{2} \)
67 \( 1 + 2.81e4T + 1.35e9T^{2} \)
71 \( 1 + 6.44e4T + 1.80e9T^{2} \)
73 \( 1 - 6.21e4T + 2.07e9T^{2} \)
79 \( 1 - 1.03e5T + 3.07e9T^{2} \)
83 \( 1 - 8.00e4T + 3.93e9T^{2} \)
89 \( 1 - 243.T + 5.58e9T^{2} \)
97 \( 1 + 1.74e5T + 8.58e9T^{2} \)
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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

−8.856757700722954867888334019592, −7.78931022341129502125149141301, −6.73821997751538254327911347938, −6.19693642046277075925641682519, −5.23658640775220291434332571294, −4.02055882232573913858352017958, −3.54271431612027466989090041896, −2.89437229024764373984432258586, −1.78759664498048685052763108768, 0, 1.78759664498048685052763108768, 2.89437229024764373984432258586, 3.54271431612027466989090041896, 4.02055882232573913858352017958, 5.23658640775220291434332571294, 6.19693642046277075925641682519, 6.73821997751538254327911347938, 7.78931022341129502125149141301, 8.856757700722954867888334019592

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