Properties

Label 2-1045-209.208-c1-0-2
Degree $2$
Conductor $1045$
Sign $-0.996 + 0.0888i$
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  + 1.53·2-s + 1.36i·3-s + 0.356·4-s − 5-s + 2.09i·6-s − 1.36i·7-s − 2.52·8-s + 1.13·9-s − 1.53·10-s + (−3.14 − 1.05i)11-s + 0.487i·12-s − 5.07·13-s − 2.10i·14-s − 1.36i·15-s − 4.58·16-s + 7.63i·17-s + ⋯
L(s)  = 1  + 1.08·2-s + 0.788i·3-s + 0.178·4-s − 0.447·5-s + 0.856i·6-s − 0.517i·7-s − 0.891·8-s + 0.378·9-s − 0.485·10-s + (−0.947 − 0.319i)11-s + 0.140i·12-s − 1.40·13-s − 0.561i·14-s − 0.352i·15-s − 1.14·16-s + 1.85i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.996 + 0.0888i$
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: \((2,\ 1045,\ (\ :1/2),\ -0.996 + 0.0888i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5117793378\)
\(L(\frac12)\) \(\approx\) \(0.5117793378\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.14 + 1.05i)T \)
19 \( 1 + (3.99 - 1.75i)T \)
good2 \( 1 - 1.53T + 2T^{2} \)
3 \( 1 - 1.36iT - 3T^{2} \)
7 \( 1 + 1.36iT - 7T^{2} \)
13 \( 1 + 5.07T + 13T^{2} \)
17 \( 1 - 7.63iT - 17T^{2} \)
23 \( 1 + 1.64T + 23T^{2} \)
29 \( 1 + 1.47T + 29T^{2} \)
31 \( 1 + 4.43iT - 31T^{2} \)
37 \( 1 + 2.89iT - 37T^{2} \)
41 \( 1 + 4.91T + 41T^{2} \)
43 \( 1 + 0.286iT - 43T^{2} \)
47 \( 1 - 4.33T + 47T^{2} \)
53 \( 1 + 12.4iT - 53T^{2} \)
59 \( 1 - 7.90iT - 59T^{2} \)
61 \( 1 + 15.1iT - 61T^{2} \)
67 \( 1 - 4.88iT - 67T^{2} \)
71 \( 1 - 7.90iT - 71T^{2} \)
73 \( 1 - 10.4iT - 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 - 1.19iT - 83T^{2} \)
89 \( 1 - 8.37iT - 89T^{2} \)
97 \( 1 - 13.3iT - 97T^{2} \)
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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

−10.33289740943958237816761979030, −9.805902949851798782050166442828, −8.643539444465892535587462657292, −7.85531374009245698131845013850, −6.83376155307386673572458898249, −5.74190284612849779514183193446, −4.96368912778692703353485363460, −4.08302198439148524193945529062, −3.72298080413432738742644337135, −2.32589008442761693579436246122, 0.14833461861488434656399449607, 2.30176393015411658447220762789, 2.97439358325405112134486380916, 4.52666212784085143001397307470, 4.90430398088451283667586909231, 5.92925761907632930633340538414, 7.09076890685702463940814022807, 7.42394371171982963909192778163, 8.591637301167473880532677884060, 9.469024720409533013530319200050

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