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Properties

Label 2-1045-5.4-c1-0-37
Degree $2$
Conductor $1045$
Sign $-0.300 - 0.953i$
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  + 1.51i·2-s − 0.900i·3-s − 0.282·4-s + (0.672 + 2.13i)5-s + 1.36·6-s − 0.603i·7-s + 2.59i·8-s + 2.18·9-s + (−3.22 + 1.01i)10-s + 11-s + 0.254i·12-s − 3.06i·13-s + 0.911·14-s + (1.92 − 0.605i)15-s − 4.48·16-s + 4.08i·17-s + ⋯
L(s)  = 1  + 1.06i·2-s − 0.520i·3-s − 0.141·4-s + (0.300 + 0.953i)5-s + 0.555·6-s − 0.228i·7-s + 0.917i·8-s + 0.729·9-s + (−1.01 + 0.321i)10-s + 0.301·11-s + 0.0735i·12-s − 0.850i·13-s + 0.243·14-s + (0.495 − 0.156i)15-s − 1.12·16-s + 0.990i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.300 - 0.953i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.300 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.000532213\)
\(L(\frac12)\) \(\approx\) \(2.000532213\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.672 - 2.13i)T \)
11 \( 1 - T \)
19 \( 1 - T \)
good2 \( 1 - 1.51iT - 2T^{2} \)
3 \( 1 + 0.900iT - 3T^{2} \)
7 \( 1 + 0.603iT - 7T^{2} \)
13 \( 1 + 3.06iT - 13T^{2} \)
17 \( 1 - 4.08iT - 17T^{2} \)
23 \( 1 - 6.19iT - 23T^{2} \)
29 \( 1 - 2.92T + 29T^{2} \)
31 \( 1 - 7.98T + 31T^{2} \)
37 \( 1 + 10.2iT - 37T^{2} \)
41 \( 1 + 7.31T + 41T^{2} \)
43 \( 1 - 0.0821iT - 43T^{2} \)
47 \( 1 - 2.77iT - 47T^{2} \)
53 \( 1 - 8.01iT - 53T^{2} \)
59 \( 1 + 3.21T + 59T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 - 3.52iT - 67T^{2} \)
71 \( 1 - 1.51T + 71T^{2} \)
73 \( 1 + 3.04iT - 73T^{2} \)
79 \( 1 - 3.57T + 79T^{2} \)
83 \( 1 - 9.37iT - 83T^{2} \)
89 \( 1 - 0.629T + 89T^{2} \)
97 \( 1 - 5.71iT - 97T^{2} \)
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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

−10.26229371305372780617890072551, −9.221119574707100186870474282653, −7.998234683276602233019677043132, −7.58070083529262397799054906478, −6.80017439495815618674097919791, −6.18672743222718782176168464742, −5.42115129650193254622882680929, −4.05396563052687218386488630213, −2.80775624004346994479154860546, −1.57779847245114694837507273132, 0.991159982373842165233647102443, 2.06536361487347697237193484926, 3.23653615914377161009913251024, 4.48392417879461310362147422443, 4.77319748397376853061938717529, 6.31224263651384769231577413023, 7.03011810805518554672322009461, 8.398844014001574746807706679830, 9.151155503837112207011405872671, 9.896443089322426393556674965671

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