Properties

Label 1-927-927.439-r1-0-0
Degree $1$
Conductor $927$
Sign $0.632 + 0.774i$
Analytic cond. $99.6199$
Root an. cond. $99.6199$
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  + (0.650 + 0.759i)2-s + (−0.153 + 0.988i)4-s + (−0.992 + 0.122i)5-s + (−0.0307 + 0.999i)7-s + (−0.850 + 0.526i)8-s + (−0.739 − 0.673i)10-s + (−0.650 − 0.759i)11-s + (0.881 − 0.473i)13-s + (−0.779 + 0.626i)14-s + (−0.952 − 0.303i)16-s + (−0.273 + 0.961i)17-s + (0.0922 − 0.995i)19-s + (0.0307 − 0.999i)20-s + (0.153 − 0.988i)22-s + (0.650 − 0.759i)23-s + ⋯
L(s)  = 1  + (0.650 + 0.759i)2-s + (−0.153 + 0.988i)4-s + (−0.992 + 0.122i)5-s + (−0.0307 + 0.999i)7-s + (−0.850 + 0.526i)8-s + (−0.739 − 0.673i)10-s + (−0.650 − 0.759i)11-s + (0.881 − 0.473i)13-s + (−0.779 + 0.626i)14-s + (−0.952 − 0.303i)16-s + (−0.273 + 0.961i)17-s + (0.0922 − 0.995i)19-s + (0.0307 − 0.999i)20-s + (0.153 − 0.988i)22-s + (0.650 − 0.759i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.632 + 0.774i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (439, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.632 + 0.774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.653390074 + 0.7842216764i\)
\(L(\frac12)\) \(\approx\) \(1.653390074 + 0.7842216764i\)
\(L(1)\) \(\approx\) \(1.007707833 + 0.5549678293i\)
\(L(1)\) \(\approx\) \(1.007707833 + 0.5549678293i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.650 + 0.759i)T \)
5 \( 1 + (-0.992 + 0.122i)T \)
7 \( 1 + (-0.0307 + 0.999i)T \)
11 \( 1 + (-0.650 - 0.759i)T \)
13 \( 1 + (0.881 - 0.473i)T \)
17 \( 1 + (-0.273 + 0.961i)T \)
19 \( 1 + (0.0922 - 0.995i)T \)
23 \( 1 + (0.650 - 0.759i)T \)
29 \( 1 + (-0.389 + 0.920i)T \)
31 \( 1 + (-0.213 - 0.976i)T \)
37 \( 1 + (-0.445 - 0.895i)T \)
41 \( 1 + (-0.389 - 0.920i)T \)
43 \( 1 + (0.998 - 0.0615i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.0922 + 0.995i)T \)
59 \( 1 + (-0.0307 - 0.999i)T \)
61 \( 1 + (0.969 - 0.243i)T \)
67 \( 1 + (0.0307 + 0.999i)T \)
71 \( 1 + (0.602 - 0.798i)T \)
73 \( 1 + (0.602 - 0.798i)T \)
79 \( 1 + (-0.389 + 0.920i)T \)
83 \( 1 + (-0.0307 + 0.999i)T \)
89 \( 1 + (-0.932 - 0.361i)T \)
97 \( 1 + (-0.696 + 0.717i)T \)
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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

−21.32041553764882718112924385093, −20.604865031734934854321875355082, −20.259626627802576565332179360608, −19.338147703381650367962046297039, −18.66184513087803209127694425864, −17.84026850080766328138935846616, −16.60432867189344356402871305506, −15.821173997615175121842262379830, −15.15510644159962512222359342464, −14.168234903401753025856908757415, −13.44628118123786719505976621498, −12.74086997759408537001752053569, −11.76820647543848936890777705149, −11.22713684389485349189174242755, −10.372172598412309498614042904371, −9.598603710039385476637749157727, −8.46389082838668685223424126385, −7.40281557602872910752729315318, −6.7134187349664504728917433482, −5.371008176849650577912051699665, −4.52603645137920104892312338777, −3.829492311201647040506208696009, −3.05356582412148258290192853908, −1.68276713414220815527563830299, −0.71287021959466615632537199388, 0.46565074772864931837091879983, 2.46255408950687081144548049670, 3.278503753340506123198566344576, 4.1084615336416801526660540386, 5.21036843094727712539048903170, 5.890572632592433818763257167219, 6.82118661398514603937704238996, 7.79564966812130674354999164653, 8.55502999402372761908504801894, 9.00694154732521632191136449431, 10.92635680303284795855797196333, 11.18526217089600288967438110350, 12.54658398391494840254347499284, 12.75403418443821858051011077497, 13.85841661292815910133035133634, 14.84087866753965232683271119183, 15.535576638252519860900385695623, 15.83466926112751912455348682477, 16.7665947549491843455413274478, 17.81563060623937639372178274993, 18.59349867526164763527433827774, 19.23357570032992431163895913619, 20.44225255163781227249825856691, 21.112636626701949975540478068578, 22.09612626977878901697157781537

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