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Properties

Label 2-927-927.77-c1-0-21
Degree $2$
Conductor $927$
Sign $-0.854 - 0.520i$
Analytic cond. $7.40213$
Root an. cond. $2.72068$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.621 + 0.263i)2-s + (1.68 + 0.415i)3-s + (−1.07 + 1.10i)4-s + (1.25 − 0.885i)5-s + (−1.15 + 0.184i)6-s + (−4.15 + 1.60i)7-s + (0.864 − 2.23i)8-s + (2.65 + 1.39i)9-s + (−0.544 + 0.879i)10-s + (0.491 − 0.650i)11-s + (−2.26 + 1.41i)12-s + (−0.978 + 6.30i)13-s + (2.15 − 2.09i)14-s + (2.47 − 0.969i)15-s + (−0.0454 − 1.47i)16-s + (0.691 + 0.243i)17-s + ⋯
L(s)  = 1  + (−0.439 + 0.185i)2-s + (0.970 + 0.239i)3-s + (−0.537 + 0.554i)4-s + (0.559 − 0.395i)5-s + (−0.471 + 0.0752i)6-s + (−1.56 + 0.607i)7-s + (0.305 − 0.788i)8-s + (0.885 + 0.465i)9-s + (−0.172 + 0.278i)10-s + (0.148 − 0.196i)11-s + (−0.654 + 0.409i)12-s + (−0.271 + 1.74i)13-s + (0.576 − 0.558i)14-s + (0.638 − 0.250i)15-s + (−0.0113 − 0.368i)16-s + (0.167 + 0.0591i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 - 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.854 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.854 - 0.520i$
Analytic conductor: \(7.40213\)
Root analytic conductor: \(2.72068\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 927,\ (\ :1/2),\ -0.854 - 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.257028 + 0.916360i\)
\(L(\frac12)\) \(\approx\) \(0.257028 + 0.916360i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.68 - 0.415i)T \)
103 \( 1 + (9.99 + 1.78i)T \)
good2 \( 1 + (0.621 - 0.263i)T + (1.39 - 1.43i)T^{2} \)
5 \( 1 + (-1.25 + 0.885i)T + (1.66 - 4.71i)T^{2} \)
7 \( 1 + (4.15 - 1.60i)T + (5.17 - 4.71i)T^{2} \)
11 \( 1 + (-0.491 + 0.650i)T + (-3.01 - 10.5i)T^{2} \)
13 \( 1 + (0.978 - 6.30i)T + (-12.3 - 3.94i)T^{2} \)
17 \( 1 + (-0.691 - 0.243i)T + (13.2 + 10.6i)T^{2} \)
19 \( 1 + (1.61 - 0.0994i)T + (18.8 - 2.33i)T^{2} \)
23 \( 1 + (0.822 + 6.64i)T + (-22.3 + 5.61i)T^{2} \)
29 \( 1 + (4.12 - 8.96i)T + (-18.8 - 22.0i)T^{2} \)
31 \( 1 + (8.71 - 0.268i)T + (30.9 - 1.90i)T^{2} \)
37 \( 1 + (-1.26 - 1.39i)T + (-3.41 + 36.8i)T^{2} \)
41 \( 1 + (-4.27 - 0.396i)T + (40.3 + 7.53i)T^{2} \)
43 \( 1 + (-2.16 - 6.81i)T + (-35.0 + 24.8i)T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 + (4.96 + 7.49i)T + (-20.6 + 48.8i)T^{2} \)
59 \( 1 + (0.940 - 2.42i)T + (-43.6 - 39.7i)T^{2} \)
61 \( 1 + (4.01 + 4.68i)T + (-9.35 + 60.2i)T^{2} \)
67 \( 1 + (-4.94 + 0.768i)T + (63.8 - 20.3i)T^{2} \)
71 \( 1 + (-6.39 - 4.53i)T + (23.5 + 66.9i)T^{2} \)
73 \( 1 + (-5.98 - 0.554i)T + (71.7 + 13.4i)T^{2} \)
79 \( 1 + (-1.56 + 1.10i)T + (26.2 - 74.5i)T^{2} \)
83 \( 1 + (-13.5 - 2.09i)T + (79.0 + 25.1i)T^{2} \)
89 \( 1 + (1.60 - 5.62i)T + (-75.6 - 46.8i)T^{2} \)
97 \( 1 + (8.58 - 10.0i)T + (-14.8 - 95.8i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.753862465990099987645306871670, −9.374899494009531435064774978844, −9.044540206347286751818750652998, −8.210609336626016871036254830726, −7.02456067410604425007645845887, −6.46075414887244848811492825608, −5.01716162259911449604348309070, −3.94888389512891916380423104884, −3.17032387627826809495415197417, −1.90584840484907786641178289038, 0.44570009966146904874578131556, 2.03787914775547856968013226578, 3.16362917300304087122867938269, 4.00784046474839767674715312603, 5.56620346622091788879068746518, 6.28718465392389897655286116199, 7.42805917417953475022390377428, 8.021146507132976547469329346217, 9.328320241618239218923272235195, 9.619281929558845510786404435677

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