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Properties

Label 1-775-775.114-r1-0-0
Degree $1$
Conductor $775$
Sign $-0.943 - 0.331i$
Analytic cond. $83.2853$
Root an. cond. $83.2853$
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  − 2-s + (0.913 + 0.406i)3-s + 4-s + (−0.913 − 0.406i)6-s + (−0.913 − 0.406i)7-s − 8-s + (0.669 + 0.743i)9-s + (0.978 + 0.207i)11-s + (0.913 + 0.406i)12-s + (−0.5 + 0.866i)13-s + (0.913 + 0.406i)14-s + 16-s + (−0.5 − 0.866i)17-s + (−0.669 − 0.743i)18-s + (−0.104 + 0.994i)19-s + ⋯
L(s)  = 1  − 2-s + (0.913 + 0.406i)3-s + 4-s + (−0.913 − 0.406i)6-s + (−0.913 − 0.406i)7-s − 8-s + (0.669 + 0.743i)9-s + (0.978 + 0.207i)11-s + (0.913 + 0.406i)12-s + (−0.5 + 0.866i)13-s + (0.913 + 0.406i)14-s + 16-s + (−0.5 − 0.866i)17-s + (−0.669 − 0.743i)18-s + (−0.104 + 0.994i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 - 0.331i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(775\)    =    \(5^{2} \cdot 31\)
Sign: $-0.943 - 0.331i$
Analytic conductor: \(83.2853\)
Root analytic conductor: \(83.2853\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{775} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 775,\ (1:\ ),\ -0.943 - 0.331i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04419140167 + 0.2589632824i\)
\(L(\frac12)\) \(\approx\) \(-0.04419140167 + 0.2589632824i\)
\(L(1)\) \(\approx\) \(0.7357596383 + 0.1655886616i\)
\(L(1)\) \(\approx\) \(0.7357596383 + 0.1655886616i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
31 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.913 + 0.406i)T \)
7 \( 1 + (-0.913 - 0.406i)T \)
11 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
41 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (0.669 - 0.743i)T \)
47 \( 1 + (0.809 - 0.587i)T \)
53 \( 1 + (-0.978 - 0.207i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (0.104 + 0.994i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (-0.978 - 0.207i)T \)
79 \( 1 + (-0.669 + 0.743i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 - T \)
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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

−21.67847923150177504228543863246, −20.45894076391866876483303694291, −19.82255681180427096323181935436, −19.39716202412836832396930540120, −18.667279033180138377720986601482, −17.75007117669579069290634370530, −17.05570892782197112886351922916, −15.98505382488172042638564340244, −15.243122879310395557132102249609, −14.662519588164651027782153635771, −13.39703290852452448867845242207, −12.61290571198350033838963319442, −11.880488881076528700263089911518, −10.711418899935197298948242661097, −9.68558767621286236075427727198, −9.23033431899530885713627366545, −8.34355993738032691257473807141, −7.62040860325072873226108969435, −6.495937461274106988869064021803, −6.12470855990896860235525746215, −4.19137274743543134241214391916, −3.03311621443764955775434670654, −2.42538816593956771472319631193, −1.224932630154326914242406837595, −0.071088818685518378791149583315, 1.4805650358886723680072011138, 2.3825483085050481527313912242, 3.49021245392869653004419998209, 4.2367070136881877245474107292, 5.85619119444105301834448667680, 7.03649066010713471752144280385, 7.38268890237105375591973072269, 8.68944961905254134579224176592, 9.32570425366658549391060043264, 9.82924120561112494801917808290, 10.69999169703952087991973754460, 11.8076227900075744417646451718, 12.632395869981696178529371379162, 13.91094037671358752357508349575, 14.46190177446245649164360005762, 15.51351812066323172130775848592, 16.27369352804685074577420647244, 16.74044058039427663458934196917, 17.79489415147439309569192754469, 18.92963331839388456738399355774, 19.292356450623755476968908159728, 20.184377180506558908257761860023, 20.5175507525924487897092637130, 21.74289199457149129201619702790, 22.28531934620900725550935775287

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