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Properties

Label 2-1045-5.4-c1-0-31
Degree $2$
Conductor $1045$
Sign $-0.465 - 0.884i$
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  + 2.58i·2-s + 0.787i·3-s − 4.66·4-s + (−1.04 − 1.97i)5-s − 2.03·6-s − 0.906i·7-s − 6.87i·8-s + 2.37·9-s + (5.10 − 2.68i)10-s − 11-s − 3.67i·12-s − 5.15i·13-s + 2.34·14-s + (1.55 − 0.820i)15-s + 8.43·16-s + 5.89i·17-s + ⋯
L(s)  = 1  + 1.82i·2-s + 0.454i·3-s − 2.33·4-s + (−0.465 − 0.884i)5-s − 0.830·6-s − 0.342i·7-s − 2.43i·8-s + 0.793·9-s + (1.61 − 0.850i)10-s − 0.301·11-s − 1.06i·12-s − 1.42i·13-s + 0.625·14-s + (0.402 − 0.211i)15-s + 2.10·16-s + 1.42i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.465 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.465 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.465 - 0.884i$
Analytic conductor: \(8.34436\)
Root analytic conductor: \(2.88866\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1045,\ (\ :1/2),\ -0.465 - 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.284614919\)
\(L(\frac12)\) \(\approx\) \(1.284614919\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.04 + 1.97i)T \)
11 \( 1 + T \)
19 \( 1 - T \)
good2 \( 1 - 2.58iT - 2T^{2} \)
3 \( 1 - 0.787iT - 3T^{2} \)
7 \( 1 + 0.906iT - 7T^{2} \)
13 \( 1 + 5.15iT - 13T^{2} \)
17 \( 1 - 5.89iT - 17T^{2} \)
23 \( 1 - 4.56iT - 23T^{2} \)
29 \( 1 - 6.53T + 29T^{2} \)
31 \( 1 - 9.15T + 31T^{2} \)
37 \( 1 - 4.63iT - 37T^{2} \)
41 \( 1 - 7.77T + 41T^{2} \)
43 \( 1 + 11.8iT - 43T^{2} \)
47 \( 1 - 2.51iT - 47T^{2} \)
53 \( 1 + 7.78iT - 53T^{2} \)
59 \( 1 + 7.54T + 59T^{2} \)
61 \( 1 - 9.66T + 61T^{2} \)
67 \( 1 + 10.0iT - 67T^{2} \)
71 \( 1 + 3.17T + 71T^{2} \)
73 \( 1 - 12.8iT - 73T^{2} \)
79 \( 1 - 4.51T + 79T^{2} \)
83 \( 1 + 8.89iT - 83T^{2} \)
89 \( 1 - 5.67T + 89T^{2} \)
97 \( 1 + 11.7iT - 97T^{2} \)
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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

−9.999351330319502759332017313195, −9.044016775241334403457552689132, −8.139969727082451966626551275985, −7.88515976636666406654347635490, −6.91033145048753251931819500061, −5.87965999675533356439815160141, −5.13610097609985766822301240904, −4.39941484795164446598294771748, −3.59864279764813231921614980452, −0.892118711069270118324452746784, 0.921106218395312671625327140200, 2.35866019182153196891640878946, 2.83747034174278413711933817203, 4.21206188258268757981324695252, 4.67259263022857087531886144946, 6.34909827359814593951927399850, 7.20208694258182455071241915118, 8.190620958426530453717303757546, 9.220911796960212556400639610352, 9.842940861401993269318199319511

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