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Properties

Label 2-1040-13.9-c1-0-21
Degree $2$
Conductor $1040$
Sign $0.477 + 0.878i$
Analytic cond. $8.30444$
Root an. cond. $2.88174$
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  + (0.5 − 0.866i)3-s + 5-s + (−0.5 − 0.866i)7-s + (1 + 1.73i)9-s + (1.5 − 2.59i)11-s + (1 − 3.46i)13-s + (0.5 − 0.866i)15-s + (1.5 + 2.59i)17-s + (−3.5 − 6.06i)19-s − 0.999·21-s + (−1.5 + 2.59i)23-s + 25-s + 5·27-s + (−1.5 + 2.59i)29-s + 4·31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + 0.447·5-s + (−0.188 − 0.327i)7-s + (0.333 + 0.577i)9-s + (0.452 − 0.783i)11-s + (0.277 − 0.960i)13-s + (0.129 − 0.223i)15-s + (0.363 + 0.630i)17-s + (−0.802 − 1.39i)19-s − 0.218·21-s + (−0.312 + 0.541i)23-s + 0.200·25-s + 0.962·27-s + (−0.278 + 0.482i)29-s + 0.718·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $0.477 + 0.878i$
Analytic conductor: \(8.30444\)
Root analytic conductor: \(2.88174\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1040,\ (\ :1/2),\ 0.477 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.961434726\)
\(L(\frac12)\) \(\approx\) \(1.961434726\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
13 \( 1 + (-1 + 3.46i)T \)
good3 \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.5 - 9.52i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.5 + 2.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.5 - 6.06i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.756191588287871648859165811934, −8.877375824938135886173928856042, −8.097190056652044338575577282229, −7.34445580816877635323052172208, −6.39531123153605777100588934769, −5.64619761202281905132792913179, −4.49503836159037450786749216550, −3.34033797054344051392410717312, −2.26446068927483415855955850151, −0.953873267213028662058595664919, 1.51152303291633495478239196345, 2.73774811435100108151228557820, 4.01923353770038868914319732041, 4.56191153329794976121191230944, 5.99986547989200094985342036163, 6.50913462040832063426356666381, 7.58104929213287439766592975022, 8.658795893503019725909953996342, 9.380871773551121011846167388122, 9.874372311729542552209458190720

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