Properties

Label 2-1045-5.4-c1-0-75
Degree $2$
Conductor $1045$
Sign $-0.783 - 0.620i$
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  − 2.10i·2-s − 1.90i·3-s − 2.43·4-s + (1.75 + 1.38i)5-s − 4.00·6-s + 0.116i·7-s + 0.913i·8-s − 0.615·9-s + (2.92 − 3.69i)10-s − 11-s + 4.62i·12-s − 2.04i·13-s + 0.246·14-s + (2.63 − 3.33i)15-s − 2.94·16-s − 6.68i·17-s + ⋯
L(s)  = 1  − 1.48i·2-s − 1.09i·3-s − 1.21·4-s + (0.783 + 0.620i)5-s − 1.63·6-s + 0.0441i·7-s + 0.322i·8-s − 0.205·9-s + (0.924 − 1.16i)10-s − 0.301·11-s + 1.33i·12-s − 0.567i·13-s + 0.0657·14-s + (0.681 − 0.860i)15-s − 0.736·16-s − 1.62i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 - 0.620i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.783 - 0.620i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.783 - 0.620i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.549319702\)
\(L(\frac12)\) \(\approx\) \(1.549319702\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.75 - 1.38i)T \)
11 \( 1 + T \)
19 \( 1 + T \)
good2 \( 1 + 2.10iT - 2T^{2} \)
3 \( 1 + 1.90iT - 3T^{2} \)
7 \( 1 - 0.116iT - 7T^{2} \)
13 \( 1 + 2.04iT - 13T^{2} \)
17 \( 1 + 6.68iT - 17T^{2} \)
23 \( 1 + 0.393iT - 23T^{2} \)
29 \( 1 + 7.51T + 29T^{2} \)
31 \( 1 - 0.564T + 31T^{2} \)
37 \( 1 + 7.53iT - 37T^{2} \)
41 \( 1 + 4.20T + 41T^{2} \)
43 \( 1 + 2.69iT - 43T^{2} \)
47 \( 1 - 13.2iT - 47T^{2} \)
53 \( 1 + 10.7iT - 53T^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 + 7.51T + 61T^{2} \)
67 \( 1 - 4.18iT - 67T^{2} \)
71 \( 1 - 1.58T + 71T^{2} \)
73 \( 1 + 0.494iT - 73T^{2} \)
79 \( 1 - 13.3T + 79T^{2} \)
83 \( 1 + 2.11iT - 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 + 10.3iT - 97T^{2} \)
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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

−9.670082441762555945553685645724, −8.933948804414579477899065837052, −7.57888768486290060055374335267, −7.04057158362707161398821521197, −6.02167566563128715468913517128, −4.97537758226077994930154021337, −3.55711827072259615804338939459, −2.55545439615531529291682998324, −1.97204766350480311551684938470, −0.68585100339320585390567118836, 1.91641822374129751858341839473, 3.79408300276939846175958784529, 4.62381543608843741901469030010, 5.37058905911035458048787823332, 6.07374469333326601225010825668, 6.92437582597120393079902217117, 8.030325823400093357872308162660, 8.750212147267396093222911084312, 9.350938443961441208013208628240, 10.18014590496560498456925690704

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