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Properties

Label 2-1045-209.208-c1-0-68
Degree $2$
Conductor $1045$
Sign $-0.861 - 0.507i$
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  − 2.04·2-s − 0.838i·3-s + 2.17·4-s − 5-s + 1.71i·6-s − 4.78i·7-s − 0.358·8-s + 2.29·9-s + 2.04·10-s + (−2.69 − 1.93i)11-s − 1.82i·12-s − 4.24·13-s + 9.76i·14-s + 0.838i·15-s − 3.61·16-s − 3.99i·17-s + ⋯
L(s)  = 1  − 1.44·2-s − 0.484i·3-s + 1.08·4-s − 0.447·5-s + 0.699i·6-s − 1.80i·7-s − 0.126·8-s + 0.765·9-s + 0.646·10-s + (−0.812 − 0.582i)11-s − 0.526i·12-s − 1.17·13-s + 2.61i·14-s + 0.216i·15-s − 0.904·16-s − 0.969i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.861 - 0.507i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.861 - 0.507i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3006935893\)
\(L(\frac12)\) \(\approx\) \(0.3006935893\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.69 + 1.93i)T \)
19 \( 1 + (4.34 - 0.391i)T \)
good2 \( 1 + 2.04T + 2T^{2} \)
3 \( 1 + 0.838iT - 3T^{2} \)
7 \( 1 + 4.78iT - 7T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 + 3.99iT - 17T^{2} \)
23 \( 1 - 5.80T + 23T^{2} \)
29 \( 1 - 0.280T + 29T^{2} \)
31 \( 1 - 1.69iT - 31T^{2} \)
37 \( 1 + 2.87iT - 37T^{2} \)
41 \( 1 - 5.74T + 41T^{2} \)
43 \( 1 - 5.09iT - 43T^{2} \)
47 \( 1 + 1.73T + 47T^{2} \)
53 \( 1 + 10.5iT - 53T^{2} \)
59 \( 1 + 1.61iT - 59T^{2} \)
61 \( 1 + 1.50iT - 61T^{2} \)
67 \( 1 - 12.0iT - 67T^{2} \)
71 \( 1 - 12.3iT - 71T^{2} \)
73 \( 1 + 0.489iT - 73T^{2} \)
79 \( 1 + 6.76T + 79T^{2} \)
83 \( 1 - 14.7iT - 83T^{2} \)
89 \( 1 - 4.01iT - 89T^{2} \)
97 \( 1 + 11.2iT - 97T^{2} \)
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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

−9.622969383613741746993661511198, −8.427875456683293603274186814345, −7.74195668053087105404208020425, −7.13972661331624213569949882119, −6.82036045628542123817732736327, −4.93657763497144965937532242590, −4.12234377654207522426251608880, −2.61954201213185867680476789432, −1.13866884963074605698514087172, −0.25029558382244635246954392906, 1.86569218368797798265596923741, 2.73388670910071034025060208831, 4.43873460051402875303214172720, 5.15760004334226280789407715772, 6.43086019809958323048618453048, 7.44218657289156693680145288837, 8.065293383910522306692607043635, 9.020188719068362309672260986746, 9.321851010687499551032853586190, 10.32498873612629690671433425923

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