Properties

Label 2-1045-1.1-c3-0-177
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  + 5.13·2-s + 1.57·3-s + 18.3·4-s − 5·5-s + 8.07·6-s − 16.5·7-s + 52.9·8-s − 24.5·9-s − 25.6·10-s − 11·11-s + 28.8·12-s − 24.1·13-s − 84.9·14-s − 7.87·15-s + 125.·16-s − 68.6·17-s − 125.·18-s + 19·19-s − 91.6·20-s − 26.0·21-s − 56.4·22-s − 112.·23-s + 83.3·24-s + 25·25-s − 123.·26-s − 81.1·27-s − 303.·28-s + ⋯
L(s)  = 1  + 1.81·2-s + 0.303·3-s + 2.29·4-s − 0.447·5-s + 0.549·6-s − 0.893·7-s + 2.33·8-s − 0.908·9-s − 0.811·10-s − 0.301·11-s + 0.694·12-s − 0.515·13-s − 1.62·14-s − 0.135·15-s + 1.95·16-s − 0.979·17-s − 1.64·18-s + 0.229·19-s − 1.02·20-s − 0.270·21-s − 0.546·22-s − 1.02·23-s + 0.709·24-s + 0.200·25-s − 0.934·26-s − 0.578·27-s − 2.04·28-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 \( 1 - 5.13T + 8T^{2} \)
3 \( 1 - 1.57T + 27T^{2} \)
7 \( 1 + 16.5T + 343T^{2} \)
13 \( 1 + 24.1T + 2.19e3T^{2} \)
17 \( 1 + 68.6T + 4.91e3T^{2} \)
23 \( 1 + 112.T + 1.21e4T^{2} \)
29 \( 1 - 97.6T + 2.43e4T^{2} \)
31 \( 1 + 33.3T + 2.97e4T^{2} \)
37 \( 1 + 133.T + 5.06e4T^{2} \)
41 \( 1 + 244.T + 6.89e4T^{2} \)
43 \( 1 - 383.T + 7.95e4T^{2} \)
47 \( 1 + 70.4T + 1.03e5T^{2} \)
53 \( 1 - 281.T + 1.48e5T^{2} \)
59 \( 1 - 32.3T + 2.05e5T^{2} \)
61 \( 1 + 411.T + 2.26e5T^{2} \)
67 \( 1 + 202.T + 3.00e5T^{2} \)
71 \( 1 - 616.T + 3.57e5T^{2} \)
73 \( 1 - 12.4T + 3.89e5T^{2} \)
79 \( 1 + 3.23T + 4.93e5T^{2} \)
83 \( 1 + 304.T + 5.71e5T^{2} \)
89 \( 1 + 706.T + 7.04e5T^{2} \)
97 \( 1 - 1.80e3T + 9.12e5T^{2} \)
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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

−9.092782300723433228927689422548, −8.057169301650715598545333710267, −7.10486739554162870137751509653, −6.35207856663946793808902643460, −5.58286077355062778823452607784, −4.65308010758475032416737628120, −3.74194653164096389622516019848, −2.97315490929309249778350323541, −2.20461141694695723318269882231, 0, 2.20461141694695723318269882231, 2.97315490929309249778350323541, 3.74194653164096389622516019848, 4.65308010758475032416737628120, 5.58286077355062778823452607784, 6.35207856663946793808902643460, 7.10486739554162870137751509653, 8.057169301650715598545333710267, 9.092782300723433228927689422548

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