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Properties

Label 1-945-945.527-r0-0-0
Degree $1$
Conductor $945$
Sign $-0.751 + 0.660i$
Analytic cond. $4.38856$
Root an. cond. $4.38856$
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  + (0.642 − 0.766i)2-s + (−0.173 − 0.984i)4-s + (−0.866 − 0.5i)8-s + (−0.766 − 0.642i)11-s + (−0.342 + 0.939i)13-s + (−0.939 + 0.342i)16-s − i·17-s − 19-s + (−0.984 + 0.173i)22-s + (0.342 − 0.939i)23-s + (0.5 + 0.866i)26-s + (−0.939 + 0.342i)29-s + (0.173 + 0.984i)31-s + (−0.342 + 0.939i)32-s + (−0.766 − 0.642i)34-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)2-s + (−0.173 − 0.984i)4-s + (−0.866 − 0.5i)8-s + (−0.766 − 0.642i)11-s + (−0.342 + 0.939i)13-s + (−0.939 + 0.342i)16-s − i·17-s − 19-s + (−0.984 + 0.173i)22-s + (0.342 − 0.939i)23-s + (0.5 + 0.866i)26-s + (−0.939 + 0.342i)29-s + (0.173 + 0.984i)31-s + (−0.342 + 0.939i)32-s + (−0.766 − 0.642i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.751 + 0.660i$
Analytic conductor: \(4.38856\)
Root analytic conductor: \(4.38856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 945,\ (0:\ ),\ -0.751 + 0.660i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2188682689 - 0.5804730216i\)
\(L(\frac12)\) \(\approx\) \(-0.2188682689 - 0.5804730216i\)
\(L(1)\) \(\approx\) \(0.8250409007 - 0.5951124928i\)
\(L(1)\) \(\approx\) \(0.8250409007 - 0.5951124928i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.642 - 0.766i)T \)
11 \( 1 + (-0.766 - 0.642i)T \)
13 \( 1 + (-0.342 + 0.939i)T \)
17 \( 1 - iT \)
19 \( 1 - T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (-0.939 + 0.342i)T \)
31 \( 1 + (0.173 + 0.984i)T \)
37 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (-0.984 - 0.173i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (-0.939 - 0.342i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (0.642 + 0.766i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.342 - 0.939i)T \)
89 \( 1 + T \)
97 \( 1 + (0.984 + 0.173i)T \)
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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

−22.537585411826573105829374684001, −21.53059027069220227198367927878, −20.97551476777675590498327027464, −20.1106122984328190355005223313, −19.105764198792933089496781964853, −18.10816370502396483453107487966, −17.31521374904152244999136455536, −16.87913047658837652044562902434, −15.61374213908023673168251631103, −15.2154795613810026989419767385, −14.581691657555587657792687628128, −13.31874195292794877255007098514, −12.96207756545754754993555918427, −12.17135348826163110155005342488, −11.07908554062054837398746919903, −10.15629566268014645501919096079, −9.129138284905627645156231177027, −8.00973776723240511064435779022, −7.6375656430157712453093150512, −6.514253689144637767282920785542, −5.68510241993858432154247495813, −4.901847418002756742914278101201, −3.96900216481051842485320140829, −2.99577575517977817649865282721, −1.93972741229361123282763883697, 0.19540177750356852217952709318, 1.69130028231572113833889989152, 2.60706390251831374224168744594, 3.47294954523635362974220892405, 4.59800054134929327282526993142, 5.19151016430971262242610643753, 6.29961378307153837506118396811, 7.08641597699796022728972845656, 8.45436073720572740528471745246, 9.229193925549366534151849938, 10.17375288139988114292241846345, 10.973394399199109062614385631437, 11.59973174785649685474148688664, 12.60102514922954927592268381516, 13.17059528280571028049562738538, 14.177355617075137402392132189560, 14.596249595523227676152717305111, 15.75515783232199376549394317029, 16.37755184661632364101151757048, 17.50782975782521979304819883228, 18.658570783479123185121567097921, 18.86424651898937669098581820620, 19.88078839214504530544190984004, 20.68644446566928657627415211900, 21.36670291394997016975464537074

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