Properties

Label 2-1045-1.1-c3-0-80
Degree $2$
Conductor $1045$
Sign $-1$
Analytic cond. $61.6569$
Root an. cond. $7.85219$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.10·2-s − 1.35·3-s − 3.57·4-s − 5·5-s + 2.84·6-s − 13.9·7-s + 24.3·8-s − 25.1·9-s + 10.5·10-s − 11·11-s + 4.82·12-s + 44.0·13-s + 29.3·14-s + 6.75·15-s − 22.6·16-s − 43.5·17-s + 52.9·18-s + 19·19-s + 17.8·20-s + 18.8·21-s + 23.1·22-s − 78.4·23-s − 32.9·24-s + 25·25-s − 92.7·26-s + 70.5·27-s + 49.8·28-s + ⋯
L(s)  = 1  − 0.743·2-s − 0.260·3-s − 0.446·4-s − 0.447·5-s + 0.193·6-s − 0.752·7-s + 1.07·8-s − 0.932·9-s + 0.332·10-s − 0.301·11-s + 0.116·12-s + 0.940·13-s + 0.559·14-s + 0.116·15-s − 0.353·16-s − 0.621·17-s + 0.693·18-s + 0.229·19-s + 0.199·20-s + 0.195·21-s + 0.224·22-s − 0.710·23-s − 0.279·24-s + 0.200·25-s − 0.699·26-s + 0.502·27-s + 0.336·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-1$
Analytic conductor: \(61.6569\)
Root analytic conductor: \(7.85219\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1045,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 5T \)
11 \( 1 + 11T \)
19 \( 1 - 19T \)
good2 \( 1 + 2.10T + 8T^{2} \)
3 \( 1 + 1.35T + 27T^{2} \)
7 \( 1 + 13.9T + 343T^{2} \)
13 \( 1 - 44.0T + 2.19e3T^{2} \)
17 \( 1 + 43.5T + 4.91e3T^{2} \)
23 \( 1 + 78.4T + 1.21e4T^{2} \)
29 \( 1 - 20.2T + 2.43e4T^{2} \)
31 \( 1 - 277.T + 2.97e4T^{2} \)
37 \( 1 - 118.T + 5.06e4T^{2} \)
41 \( 1 - 293.T + 6.89e4T^{2} \)
43 \( 1 - 171.T + 7.95e4T^{2} \)
47 \( 1 - 265.T + 1.03e5T^{2} \)
53 \( 1 + 384.T + 1.48e5T^{2} \)
59 \( 1 + 472.T + 2.05e5T^{2} \)
61 \( 1 - 617.T + 2.26e5T^{2} \)
67 \( 1 + 24.8T + 3.00e5T^{2} \)
71 \( 1 - 828.T + 3.57e5T^{2} \)
73 \( 1 + 350.T + 3.89e5T^{2} \)
79 \( 1 + 475.T + 4.93e5T^{2} \)
83 \( 1 - 757.T + 5.71e5T^{2} \)
89 \( 1 + 182.T + 7.04e5T^{2} \)
97 \( 1 - 161.T + 9.12e5T^{2} \)
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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

−9.096593866770828933225175214225, −8.347789066434480712665029457585, −7.78348640422955987533210337886, −6.57601188507842045496533285751, −5.83989285977296845573851948933, −4.69010053106848791105601472676, −3.77237616558136031655062222686, −2.62765413048744347356794725217, −0.938932031316093563205343103975, 0, 0.938932031316093563205343103975, 2.62765413048744347356794725217, 3.77237616558136031655062222686, 4.69010053106848791105601472676, 5.83989285977296845573851948933, 6.57601188507842045496533285751, 7.78348640422955987533210337886, 8.347789066434480712665029457585, 9.096593866770828933225175214225

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