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Properties

Label 2-1045-209.208-c1-0-21
Degree $2$
Conductor $1045$
Sign $0.731 - 0.682i$
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  − 2.52·2-s − 0.384i·3-s + 4.39·4-s − 5-s + 0.973i·6-s + 1.35i·7-s − 6.07·8-s + 2.85·9-s + 2.52·10-s + (1.95 + 2.68i)11-s − 1.69i·12-s + 2.33·13-s − 3.42i·14-s + 0.384i·15-s + 6.55·16-s − 5.43i·17-s + ⋯
L(s)  = 1  − 1.78·2-s − 0.222i·3-s + 2.19·4-s − 0.447·5-s + 0.397i·6-s + 0.511i·7-s − 2.14·8-s + 0.950·9-s + 0.799·10-s + (0.587 + 0.808i)11-s − 0.488i·12-s + 0.648·13-s − 0.914i·14-s + 0.0993i·15-s + 1.63·16-s − 1.31i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.731 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.731 - 0.682i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (626, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ 0.731 - 0.682i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6865231996\)
\(L(\frac12)\) \(\approx\) \(0.6865231996\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.95 - 2.68i)T \)
19 \( 1 + (-0.531 - 4.32i)T \)
good2 \( 1 + 2.52T + 2T^{2} \)
3 \( 1 + 0.384iT - 3T^{2} \)
7 \( 1 - 1.35iT - 7T^{2} \)
13 \( 1 - 2.33T + 13T^{2} \)
17 \( 1 + 5.43iT - 17T^{2} \)
23 \( 1 - 0.981T + 23T^{2} \)
29 \( 1 + 3.04T + 29T^{2} \)
31 \( 1 - 8.09iT - 31T^{2} \)
37 \( 1 + 1.40iT - 37T^{2} \)
41 \( 1 - 1.12T + 41T^{2} \)
43 \( 1 - 2.01iT - 43T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 - 3.95iT - 53T^{2} \)
59 \( 1 + 1.26iT - 59T^{2} \)
61 \( 1 + 9.55iT - 61T^{2} \)
67 \( 1 - 13.2iT - 67T^{2} \)
71 \( 1 + 14.3iT - 71T^{2} \)
73 \( 1 + 2.05iT - 73T^{2} \)
79 \( 1 - 9.57T + 79T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 - 12.3iT - 89T^{2} \)
97 \( 1 - 2.23iT - 97T^{2} \)
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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

−9.761220145682568135982723156485, −9.272339318113583696952063358552, −8.437095873742661113569833999106, −7.61071001158434735160260552908, −7.04073043453800355120884022136, −6.29631501760655001903110418145, −4.85554667683339521676243056676, −3.46274685769753919460718996676, −2.05772543983490329845075052870, −1.11670411316164085837586345609, 0.69839578849947696724726766804, 1.77162463583031093364209135956, 3.38992442146722908042364940999, 4.31622490590061406971509031176, 6.00354134037778779115127347752, 6.80299541227080057671028048450, 7.53549255470357658711941377634, 8.328257246206063776478228272945, 8.964834294132152961501190469559, 9.765113486123940352266007054217

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