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Properties

Label 1-927-927.473-r1-0-0
Degree $1$
Conductor $927$
Sign $-0.866 + 0.499i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.866 + 0.499i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (473, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ -0.866 + 0.499i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1276517092 + 0.4772527330i\)
\(L(\frac12)\) \(\approx\) \(0.1276517092 + 0.4772527330i\)
\(L(1)\) \(\approx\) \(0.5746772931 + 0.1698784761i\)
\(L(1)\) \(\approx\) \(0.5746772931 + 0.1698784761i\)

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

−21.305853217576594113841003688021, −20.41471267443005482569331628017, −19.689569431811020154943708816544, −18.93082507657729042256668883773, −18.34637498978182165824807998314, −17.78062062155058255198137689131, −16.47031088146149831704393492899, −15.78018881998613140049217451487, −15.39477552719716679196097165618, −13.882996344069922018779135725044, −13.1414529786229280172976157523, −12.18322668906629011955813527953, −11.507341563753524525135454855177, −11.03141096770332262275671206768, −9.96015209337138266095582158216, −9.00504558348453837223902872297, −8.26236786816966440210305181152, −7.739134210254347813596794306447, −6.550499817829981041573144612589, −5.33638393306190311917001193709, −4.29438073380255556135778619251, −3.01463202015323751496130096507, −2.84650017607948687162716065688, −1.1965072064638144935680524973, −0.188119258525457781153040935724, 0.86574024628815987694292009807, 1.94705815730350737810965132970, 3.76315681007977784054161715610, 4.29164088343221737553253476119, 5.4341730060842015660321260420, 6.52885233042261419075773349043, 7.271523947833975657051411635, 8.05973512967865116034509044384, 8.55538665161432544601825744127, 9.87436545152761019699987941983, 10.486971757110978884005516583051, 11.223310226857619233154343209274, 12.38987860374975197503543131344, 13.270741101061943155110390645192, 14.33053740689461604686886202321, 14.885516602609134502902477686341, 15.92897815403751101893053150973, 16.33376990428068004244989160673, 17.104865222480140417958657871904, 18.067397902119836470119126403240, 18.75808279682805184588654917561, 19.50231027060469818736471134070, 20.34372397814247798583312543820, 20.78147793614143735232161394750, 22.32719026386278948058715903068

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