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Properties

Label 2-1045-209.208-c1-0-6
Degree $2$
Conductor $1045$
Sign $0.751 - 0.659i$
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  − 2.71·2-s − 2.28i·3-s + 5.39·4-s − 5-s + 6.20i·6-s + 2.59i·7-s − 9.23·8-s − 2.21·9-s + 2.71·10-s + (−2.91 − 1.57i)11-s − 12.3i·12-s − 3.84·13-s − 7.05i·14-s + 2.28i·15-s + 14.3·16-s − 1.32i·17-s + ⋯
L(s)  = 1  − 1.92·2-s − 1.31i·3-s + 2.69·4-s − 0.447·5-s + 2.53i·6-s + 0.980i·7-s − 3.26·8-s − 0.737·9-s + 0.860·10-s + (−0.879 − 0.476i)11-s − 3.55i·12-s − 1.06·13-s − 1.88i·14-s + 0.589i·15-s + 3.58·16-s − 0.321i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.751 - 0.659i$
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: \((2,\ 1045,\ (\ :1/2),\ 0.751 - 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2707716397\)
\(L(\frac12)\) \(\approx\) \(0.2707716397\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.91 + 1.57i)T \)
19 \( 1 + (-1.51 + 4.08i)T \)
good2 \( 1 + 2.71T + 2T^{2} \)
3 \( 1 + 2.28iT - 3T^{2} \)
7 \( 1 - 2.59iT - 7T^{2} \)
13 \( 1 + 3.84T + 13T^{2} \)
17 \( 1 + 1.32iT - 17T^{2} \)
23 \( 1 + 4.75T + 23T^{2} \)
29 \( 1 - 5.07T + 29T^{2} \)
31 \( 1 - 5.94iT - 31T^{2} \)
37 \( 1 - 0.538iT - 37T^{2} \)
41 \( 1 + 6.13T + 41T^{2} \)
43 \( 1 - 10.0iT - 43T^{2} \)
47 \( 1 - 9.19T + 47T^{2} \)
53 \( 1 + 1.16iT - 53T^{2} \)
59 \( 1 - 12.3iT - 59T^{2} \)
61 \( 1 + 6.07iT - 61T^{2} \)
67 \( 1 + 5.71iT - 67T^{2} \)
71 \( 1 - 11.0iT - 71T^{2} \)
73 \( 1 - 15.5iT - 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + 7.45iT - 83T^{2} \)
89 \( 1 - 8.00iT - 89T^{2} \)
97 \( 1 - 17.6iT - 97T^{2} \)
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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

−9.865889131976488882430854185718, −8.961030984291572102365509752105, −8.271422183401744701763437969399, −7.72087790527403047086380982451, −7.03609291922386271102862213718, −6.32969994389499555445744826622, −5.25240657756347482354798546896, −2.78528109475033597046199297236, −2.35513238336157054577508377159, −0.971001730012542201682423443614, 0.27528052411734312015615331237, 2.07647994208526694567378925983, 3.38046581111123475951326455874, 4.41895483761090667256060652333, 5.65168100172156548964343941989, 6.96411586817252653702896105642, 7.63841279971318074501131996649, 8.204969969479954864799016133661, 9.239772450919557462528596570880, 10.00481690774769106788157143970

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