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Properties

Label 2-1045-209.208-c1-0-71
Degree $2$
Conductor $1045$
Sign $0.260 + 0.965i$
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  + 1.83·2-s − 0.556i·3-s + 1.36·4-s + 5-s − 1.02i·6-s − 4.08i·7-s − 1.15·8-s + 2.69·9-s + 1.83·10-s + (2.99 − 1.43i)11-s − 0.760i·12-s − 5.36·13-s − 7.50i·14-s − 0.556i·15-s − 4.86·16-s − 5.36i·17-s + ⋯
L(s)  = 1  + 1.29·2-s − 0.321i·3-s + 0.684·4-s + 0.447·5-s − 0.416i·6-s − 1.54i·7-s − 0.409·8-s + 0.896·9-s + 0.580·10-s + (0.901 − 0.432i)11-s − 0.219i·12-s − 1.48·13-s − 2.00i·14-s − 0.143i·15-s − 1.21·16-s − 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.260 + 0.965i$
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: \((2,\ 1045,\ (\ :1/2),\ 0.260 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.184670200\)
\(L(\frac12)\) \(\approx\) \(3.184670200\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.99 + 1.43i)T \)
19 \( 1 + (0.797 - 4.28i)T \)
good2 \( 1 - 1.83T + 2T^{2} \)
3 \( 1 + 0.556iT - 3T^{2} \)
7 \( 1 + 4.08iT - 7T^{2} \)
13 \( 1 + 5.36T + 13T^{2} \)
17 \( 1 + 5.36iT - 17T^{2} \)
23 \( 1 + 1.90T + 23T^{2} \)
29 \( 1 - 8.05T + 29T^{2} \)
31 \( 1 - 5.60iT - 31T^{2} \)
37 \( 1 - 3.42iT - 37T^{2} \)
41 \( 1 - 9.21T + 41T^{2} \)
43 \( 1 + 6.62iT - 43T^{2} \)
47 \( 1 - 7.01T + 47T^{2} \)
53 \( 1 + 7.74iT - 53T^{2} \)
59 \( 1 - 3.01iT - 59T^{2} \)
61 \( 1 - 7.25iT - 61T^{2} \)
67 \( 1 + 6.53iT - 67T^{2} \)
71 \( 1 + 0.908iT - 71T^{2} \)
73 \( 1 - 9.46iT - 73T^{2} \)
79 \( 1 + 9.42T + 79T^{2} \)
83 \( 1 - 3.44iT - 83T^{2} \)
89 \( 1 + 6.38iT - 89T^{2} \)
97 \( 1 - 12.5iT - 97T^{2} \)
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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

−9.969088742881435254090250446739, −9.062403358680827952963027950676, −7.67594890935869569036531554552, −6.96053605377955499861919318060, −6.42795959875392206234249678169, −5.18601356146120344470755713464, −4.43186455607918259834295895701, −3.75383025853626412823851588147, −2.52877374662886804701450093420, −0.988887527019411439073644520319, 2.05124154991885284880180029760, 2.82945849802479115484296964188, 4.27641911603150461731397347419, 4.66878286035708631776976765127, 5.74488585098025413438961705452, 6.30324995365796208812931436056, 7.29008880444080727037804828332, 8.656263584597300126263627552288, 9.387579104566490680971398065726, 9.912637172448787910435604371569

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