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Properties

Label 1-1045-1045.1033-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.960 + 0.277i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + (0.5 + 0.866i)6-s + i·7-s − i·8-s + (0.5 + 0.866i)9-s − i·12-s + (−0.866 + 0.5i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s − i·18-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s + (−0.5 + 0.866i)24-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + (0.5 + 0.866i)6-s + i·7-s − i·8-s + (0.5 + 0.866i)9-s − i·12-s + (−0.866 + 0.5i)13-s + (0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (−0.866 − 0.5i)17-s − i·18-s + (0.5 − 0.866i)21-s + (0.866 − 0.5i)23-s + (−0.5 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.960 + 0.277i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (1033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.960 + 0.277i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5443601223 + 0.07692600124i\)
\(L(\frac12)\) \(\approx\) \(0.5443601223 + 0.07692600124i\)
\(L(1)\) \(\approx\) \(0.5209297542 - 0.07689484181i\)
\(L(1)\) \(\approx\) \(0.5209297542 - 0.07689484181i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + iT \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + T \)
37 \( 1 - iT \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.866 + 0.5i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.866 - 0.5i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.47327878015452881589752130727, −20.581783213648123337091370445235, −19.8685133685515457520323737216, −19.15472347306011571220895682599, −18.00089306049226467591516025144, −17.51321481732666989581877434644, −16.8728690406222289459216247228, −16.3335938606727262892864165529, −15.2732053557127981930710584800, −14.91175097905729790987015331569, −13.710357148630060640642703735697, −12.72176033682057169388232285313, −11.604488049030045437488330489436, −10.89868546410095062070134703510, −10.26907979781919339110397972747, −9.62525140833596527223259138894, −8.67241154656377123304780020834, −7.54504738116138838518634269118, −6.915970085189501143827318184598, −6.113646670441516698440107299925, −5.09135166794655924254655013791, −4.42205260799564116160627717903, −3.08503598013890094099152326563, −1.55962597801015892087824293535, −0.50124181846524123886167446913, 0.806035427595313350855016859399, 2.18167157624989987602680470282, 2.549587825746266572681273035466, 4.198638389749082082279780556389, 5.1642812299091336014572475086, 6.25859040893796404198096949740, 6.98772543693698098014976283962, 7.798878518257466319512070861547, 8.82396007049097445116922648333, 9.51511756475562509572976579609, 10.46105518519058880568727885443, 11.41419206499502541513718566351, 11.77773501670155295319130302139, 12.64154234643899290183059575317, 13.23000285197610276301072902997, 14.57930824909759206978374381568, 15.708932041241970165301101361800, 16.220632033218360818208205497347, 17.27181624302668191327872678224, 17.63451301582789571165290619545, 18.52275827260684790263611375885, 19.11495967092968356458173908292, 19.64545879438205959532869340501, 20.94460532981227511245458638441, 21.44039207278451833784292120682

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