Properties

Label 2-1045-5.4-c1-0-28
Degree $2$
Conductor $1045$
Sign $0.873 - 0.486i$
Analytic cond. $8.34436$
Root an. cond. $2.88866$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.387i·2-s − 0.494i·3-s + 1.84·4-s + (−1.95 + 1.08i)5-s − 0.191·6-s + 2.37i·7-s − 1.49i·8-s + 2.75·9-s + (0.421 + 0.756i)10-s − 11-s − 0.914i·12-s + 0.475i·13-s + 0.919·14-s + (0.538 + 0.965i)15-s + 3.12·16-s + 0.596i·17-s + ⋯
L(s)  = 1  − 0.273i·2-s − 0.285i·3-s + 0.924·4-s + (−0.873 + 0.486i)5-s − 0.0781·6-s + 0.896i·7-s − 0.527i·8-s + 0.918·9-s + (0.133 + 0.239i)10-s − 0.301·11-s − 0.263i·12-s + 0.131i·13-s + 0.245·14-s + (0.138 + 0.249i)15-s + 0.780·16-s + 0.144i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.873 - 0.486i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.873 - 0.486i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ 0.873 - 0.486i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.776913252\)
\(L(\frac12)\) \(\approx\) \(1.776913252\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.95 - 1.08i)T \)
11 \( 1 + T \)
19 \( 1 + T \)
good2 \( 1 + 0.387iT - 2T^{2} \)
3 \( 1 + 0.494iT - 3T^{2} \)
7 \( 1 - 2.37iT - 7T^{2} \)
13 \( 1 - 0.475iT - 13T^{2} \)
17 \( 1 - 0.596iT - 17T^{2} \)
23 \( 1 - 8.79iT - 23T^{2} \)
29 \( 1 + 3.72T + 29T^{2} \)
31 \( 1 - 7.08T + 31T^{2} \)
37 \( 1 - 8.28iT - 37T^{2} \)
41 \( 1 - 6.65T + 41T^{2} \)
43 \( 1 - 10.7iT - 43T^{2} \)
47 \( 1 - 3.68iT - 47T^{2} \)
53 \( 1 + 1.59iT - 53T^{2} \)
59 \( 1 - 7.09T + 59T^{2} \)
61 \( 1 + 6.32T + 61T^{2} \)
67 \( 1 + 10.2iT - 67T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 + 12.0iT - 73T^{2} \)
79 \( 1 - 16.5T + 79T^{2} \)
83 \( 1 + 10.5iT - 83T^{2} \)
89 \( 1 + 7.65T + 89T^{2} \)
97 \( 1 + 3.30iT - 97T^{2} \)
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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

−10.10330930421270656613657770304, −9.298946717864302655270976019641, −7.953287976500856377957000668373, −7.61389246004769758928672450413, −6.64924242990514147723786501875, −5.97267764168531691631895374464, −4.64012220190335935104776737929, −3.48958313508395158225522777277, −2.61629223567932070810128818464, −1.45724806949068192598537227094, 0.861274610707743954458580422574, 2.43074445902524973203137801936, 3.82256068863662557324947540281, 4.44245320949356006298597400331, 5.53591984232612148304344443707, 6.79216562546406367838848599078, 7.27235869552280282263731440072, 8.020369774730017465238359140419, 8.862400536724500830778214463132, 10.12828810358082744605747478261

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