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Properties

Label 1-1040-1040.907-r0-0-0
Degree $1$
Conductor $1040$
Sign $-0.147 + 0.989i$
Analytic cond. $4.82973$
Root an. cond. $4.82973$
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  + (−0.5 + 0.866i)3-s + (−0.866 + 0.5i)7-s + (−0.5 − 0.866i)9-s + (−0.866 − 0.5i)11-s + (0.866 − 0.5i)17-s + (0.866 − 0.5i)19-s − i·21-s + (0.866 + 0.5i)23-s + 27-s + (−0.866 − 0.5i)29-s + 31-s + (0.866 − 0.5i)33-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.866 + 0.5i)7-s + (−0.5 − 0.866i)9-s + (−0.866 − 0.5i)11-s + (0.866 − 0.5i)17-s + (0.866 − 0.5i)19-s − i·21-s + (0.866 + 0.5i)23-s + 27-s + (−0.866 − 0.5i)29-s + 31-s + (0.866 − 0.5i)33-s + (−0.5 + 0.866i)37-s + (−0.5 + 0.866i)41-s + (0.5 + 0.866i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.147 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.147 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1040\)    =    \(2^{4} \cdot 5 \cdot 13\)
Sign: $-0.147 + 0.989i$
Analytic conductor: \(4.82973\)
Root analytic conductor: \(4.82973\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1040} (907, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1040,\ (0:\ ),\ -0.147 + 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5620397431 + 0.6520804950i\)
\(L(\frac12)\) \(\approx\) \(0.5620397431 + 0.6520804950i\)
\(L(1)\) \(\approx\) \(0.7375604653 + 0.2614970175i\)
\(L(1)\) \(\approx\) \(0.7375604653 + 0.2614970175i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (0.866 - 0.5i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 - iT \)
53 \( 1 + T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 + (-0.866 + 0.5i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 - iT \)
79 \( 1 + T \)
83 \( 1 - T \)
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

−21.313029726571942635366244241409, −20.45577257243122401430425352899, −19.69879929326175329108710385908, −18.795146125932878445470978184398, −18.4518901450215643532630533929, −17.35681197481579163349799608330, −16.78189528699453435456265612200, −16.04402832624091002982680389143, −15.09031322904698601234932581400, −13.95806960624454873798224732277, −13.39086035272423199029298281518, −12.46382239987486518323861245903, −12.17272677682570705933272268306, −10.80206683092309892637734838601, −10.35680260814100442378168772424, −9.329107057510236363686522891593, −8.14433527369957591634852018755, −7.37364142032770911441586018118, −6.79120300129881278256126299625, −5.73238142769190917838637564772, −5.10655563502148227599060987803, −3.734931437982671081694878528884, −2.78562443884078696853980941423, −1.67231417525568365460471844519, −0.496370007570122139537267574935, 0.94575591325712014100961457639, 2.92911917757923051315579453811, 3.14275183498676850146787234148, 4.51422497404114412724598152532, 5.40789508363140310922638321432, 5.942321527080478247410430555574, 7.0167791825328219601991217921, 8.09221704322228351360843029775, 9.194908326289272339689371610, 9.70093858722146634976132440800, 10.5037798466972217478641380297, 11.44224539898014974407857804724, 12.046045831791611584498261703697, 13.086697785613460886006502465115, 13.81268638081524321770755894095, 15.03060271398241875708308201824, 15.56245527390203050825349532047, 16.266232034359479164913994428185, 16.84948252784829285053827943753, 17.86465126314570028358348934893, 18.64193003615567562276117997568, 19.37101646242013328653704224718, 20.424952606979746940827927112573, 21.12832200601432737038159506682, 21.6975854625270094029009343387

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