Properties

Label 2-1045-1.1-c1-0-24
Degree $2$
Conductor $1045$
Sign $-1$
Analytic cond. $8.34436$
Root an. cond. $2.88866$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.0881·2-s − 3.20·3-s − 1.99·4-s + 5-s − 0.282·6-s − 1.95·7-s − 0.351·8-s + 7.26·9-s + 0.0881·10-s − 11-s + 6.38·12-s + 3.20·13-s − 0.172·14-s − 3.20·15-s + 3.95·16-s + 0.503·17-s + 0.640·18-s + 19-s − 1.99·20-s + 6.25·21-s − 0.0881·22-s + 1.05·23-s + 1.12·24-s + 25-s + 0.282·26-s − 13.6·27-s + 3.88·28-s + ⋯
L(s)  = 1  + 0.0623·2-s − 1.84·3-s − 0.996·4-s + 0.447·5-s − 0.115·6-s − 0.737·7-s − 0.124·8-s + 2.42·9-s + 0.0278·10-s − 0.301·11-s + 1.84·12-s + 0.889·13-s − 0.0459·14-s − 0.827·15-s + 0.988·16-s + 0.122·17-s + 0.150·18-s + 0.229·19-s − 0.445·20-s + 1.36·21-s − 0.0187·22-s + 0.220·23-s + 0.230·24-s + 0.200·25-s + 0.0554·26-s − 2.62·27-s + 0.734·28-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
11 \( 1 + T \)
19 \( 1 - T \)
good2 \( 1 - 0.0881T + 2T^{2} \)
3 \( 1 + 3.20T + 3T^{2} \)
7 \( 1 + 1.95T + 7T^{2} \)
13 \( 1 - 3.20T + 13T^{2} \)
17 \( 1 - 0.503T + 17T^{2} \)
23 \( 1 - 1.05T + 23T^{2} \)
29 \( 1 + 7.91T + 29T^{2} \)
31 \( 1 - 9.52T + 31T^{2} \)
37 \( 1 + 5.22T + 37T^{2} \)
41 \( 1 - 8.32T + 41T^{2} \)
43 \( 1 + 8.04T + 43T^{2} \)
47 \( 1 + 9.39T + 47T^{2} \)
53 \( 1 - 2.37T + 53T^{2} \)
59 \( 1 + 2.80T + 59T^{2} \)
61 \( 1 + 1.53T + 61T^{2} \)
67 \( 1 + 9.79T + 67T^{2} \)
71 \( 1 + 4.45T + 71T^{2} \)
73 \( 1 + 1.55T + 73T^{2} \)
79 \( 1 - 3.00T + 79T^{2} \)
83 \( 1 + 2.22T + 83T^{2} \)
89 \( 1 - 6.10T + 89T^{2} \)
97 \( 1 + 4.69T + 97T^{2} \)
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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

−9.808584307325280268850538388518, −8.903310233301945018355952283008, −7.70747774896139184906052044713, −6.57031995808677251812175765834, −5.99488277320995693797376354037, −5.27799974047905390366857456996, −4.50551575743991790104397101702, −3.42197831029460537931350324079, −1.28254953411861682805164698558, 0, 1.28254953411861682805164698558, 3.42197831029460537931350324079, 4.50551575743991790104397101702, 5.27799974047905390366857456996, 5.99488277320995693797376354037, 6.57031995808677251812175765834, 7.70747774896139184906052044713, 8.903310233301945018355952283008, 9.808584307325280268850538388518

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