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Properties

Label 2-1045-209.208-c1-0-13
Degree $2$
Conductor $1045$
Sign $-0.225 - 0.974i$
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  − 2.17·2-s + 1.39i·3-s + 2.73·4-s + 5-s − 3.03i·6-s − 0.972i·7-s − 1.58·8-s + 1.05·9-s − 2.17·10-s + (−1.20 − 3.08i)11-s + 3.81i·12-s − 5.65·13-s + 2.11i·14-s + 1.39i·15-s − 2.00·16-s + 4.67i·17-s + ⋯
L(s)  = 1  − 1.53·2-s + 0.806i·3-s + 1.36·4-s + 0.447·5-s − 1.23i·6-s − 0.367i·7-s − 0.561·8-s + 0.350·9-s − 0.687·10-s + (−0.364 − 0.931i)11-s + 1.10i·12-s − 1.56·13-s + 0.565i·14-s + 0.360i·15-s − 0.501·16-s + 1.13i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.225 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.225 - 0.974i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.225 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6187581127\)
\(L(\frac12)\) \(\approx\) \(0.6187581127\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.20 + 3.08i)T \)
19 \( 1 + (-4.31 - 0.631i)T \)
good2 \( 1 + 2.17T + 2T^{2} \)
3 \( 1 - 1.39iT - 3T^{2} \)
7 \( 1 + 0.972iT - 7T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 - 4.67iT - 17T^{2} \)
23 \( 1 + 1.62T + 23T^{2} \)
29 \( 1 + 6.29T + 29T^{2} \)
31 \( 1 - 8.09iT - 31T^{2} \)
37 \( 1 + 4.08iT - 37T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 - 9.06iT - 43T^{2} \)
47 \( 1 - 4.98T + 47T^{2} \)
53 \( 1 - 1.30iT - 53T^{2} \)
59 \( 1 - 3.03iT - 59T^{2} \)
61 \( 1 - 12.2iT - 61T^{2} \)
67 \( 1 - 8.46iT - 67T^{2} \)
71 \( 1 + 15.6iT - 71T^{2} \)
73 \( 1 - 3.75iT - 73T^{2} \)
79 \( 1 - 0.484T + 79T^{2} \)
83 \( 1 - 7.80iT - 83T^{2} \)
89 \( 1 + 7.55iT - 89T^{2} \)
97 \( 1 - 9.02iT - 97T^{2} \)
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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

−10.12673988197774140525200724219, −9.369378932536580252368284703456, −8.833045888409953022705354619451, −7.66119863645432349402595290529, −7.30845999129829847204589652105, −6.01004419828713868771044022893, −5.00915885012920269280321938632, −3.89145822911061755940006203717, −2.54962287226845535908852300419, −1.20333757820295512295174551926, 0.51190816031746637783074389931, 1.98210330331937968861559840675, 2.48277117233684518158709845210, 4.56226810728504889980722119908, 5.58721906190589285824414217347, 6.87997950582818308489207539552, 7.45996465836544123756400303778, 7.74054537890291751414226045667, 9.072895450959724395381593703278, 9.771500524567512601339416424192

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