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Properties

Label 2-1045-5.4-c1-0-84
Degree $2$
Conductor $1045$
Sign $-0.116 - 0.993i$
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  − 2.39i·2-s − 0.881i·3-s − 3.75·4-s + (−0.259 − 2.22i)5-s − 2.11·6-s − 2.42i·7-s + 4.20i·8-s + 2.22·9-s + (−5.32 + 0.623i)10-s + 11-s + 3.30i·12-s − 3.61i·13-s − 5.82·14-s + (−1.95 + 0.229i)15-s + 2.57·16-s − 6.97i·17-s + ⋯
L(s)  = 1  − 1.69i·2-s − 0.509i·3-s − 1.87·4-s + (−0.116 − 0.993i)5-s − 0.863·6-s − 0.917i·7-s + 1.48i·8-s + 0.740·9-s + (−1.68 + 0.197i)10-s + 0.301·11-s + 0.954i·12-s − 1.00i·13-s − 1.55·14-s + (−0.505 + 0.0591i)15-s + 0.643·16-s − 1.69i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.116 - 0.993i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.116 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.396217940\)
\(L(\frac12)\) \(\approx\) \(1.396217940\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.259 + 2.22i)T \)
11 \( 1 - T \)
19 \( 1 + T \)
good2 \( 1 + 2.39iT - 2T^{2} \)
3 \( 1 + 0.881iT - 3T^{2} \)
7 \( 1 + 2.42iT - 7T^{2} \)
13 \( 1 + 3.61iT - 13T^{2} \)
17 \( 1 + 6.97iT - 17T^{2} \)
23 \( 1 - 8.41iT - 23T^{2} \)
29 \( 1 - 5.07T + 29T^{2} \)
31 \( 1 + 0.478T + 31T^{2} \)
37 \( 1 - 3.19iT - 37T^{2} \)
41 \( 1 - 10.7T + 41T^{2} \)
43 \( 1 - 2.14iT - 43T^{2} \)
47 \( 1 + 3.37iT - 47T^{2} \)
53 \( 1 - 6.23iT - 53T^{2} \)
59 \( 1 - 1.66T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 - 6.57iT - 67T^{2} \)
71 \( 1 - 3.86T + 71T^{2} \)
73 \( 1 - 6.54iT - 73T^{2} \)
79 \( 1 + 7.07T + 79T^{2} \)
83 \( 1 - 0.498iT - 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + 9.39iT - 97T^{2} \)
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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

−9.746575598645561007298737355194, −8.776421797169352159013678523301, −7.75031039364388206253491670004, −7.06151466793989910181475389727, −5.47343939678175497780419950413, −4.53776728341831548281820813912, −3.84350176453942089075407019679, −2.69286869143386616174822968371, −1.30556053242668506147126412596, −0.73392448660813800735501249723, 2.26283486855549810167199144437, 3.93380885792875488330704730963, 4.49007287933005920929921195785, 5.72660822920796613030596117800, 6.46933613952268853093699372601, 6.88611126750374472286991990011, 7.989024955001565720399289791889, 8.680320829472067938059079805274, 9.414724130550881704493810497259, 10.32568328887310610946037033610

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