Properties

Label 2-927-927.77-c1-0-48
Degree $2$
Conductor $927$
Sign $-0.0658 - 0.997i$
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  + (−1.39 + 0.589i)2-s + (1.04 + 1.38i)3-s + (0.200 − 0.206i)4-s + (1.33 − 0.945i)5-s + (−2.26 − 1.31i)6-s + (−2.69 + 1.04i)7-s + (0.935 − 2.41i)8-s + (−0.835 + 2.88i)9-s + (−1.30 + 2.10i)10-s + (2.20 − 2.91i)11-s + (0.495 + 0.0625i)12-s + (0.508 − 3.27i)13-s + (3.13 − 3.03i)14-s + (2.69 + 0.865i)15-s + (0.138 + 4.49i)16-s + (2.84 + 1.00i)17-s + ⋯
L(s)  = 1  + (−0.985 + 0.416i)2-s + (0.600 + 0.799i)3-s + (0.100 − 0.103i)4-s + (0.597 − 0.422i)5-s + (−0.925 − 0.537i)6-s + (−1.01 + 0.394i)7-s + (0.330 − 0.853i)8-s + (−0.278 + 0.960i)9-s + (−0.412 + 0.665i)10-s + (0.663 − 0.878i)11-s + (0.143 + 0.0180i)12-s + (0.141 − 0.909i)13-s + (0.837 − 0.812i)14-s + (0.696 + 0.223i)15-s + (0.0345 + 1.12i)16-s + (0.690 + 0.243i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0658 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0658 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.0658 - 0.997i$
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: \((2,\ 927,\ (\ :1/2),\ -0.0658 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.771888 + 0.824526i\)
\(L(\frac12)\) \(\approx\) \(0.771888 + 0.824526i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.04 - 1.38i)T \)
103 \( 1 + (9.87 - 2.35i)T \)
good2 \( 1 + (1.39 - 0.589i)T + (1.39 - 1.43i)T^{2} \)
5 \( 1 + (-1.33 + 0.945i)T + (1.66 - 4.71i)T^{2} \)
7 \( 1 + (2.69 - 1.04i)T + (5.17 - 4.71i)T^{2} \)
11 \( 1 + (-2.20 + 2.91i)T + (-3.01 - 10.5i)T^{2} \)
13 \( 1 + (-0.508 + 3.27i)T + (-12.3 - 3.94i)T^{2} \)
17 \( 1 + (-2.84 - 1.00i)T + (13.2 + 10.6i)T^{2} \)
19 \( 1 + (-7.08 + 0.436i)T + (18.8 - 2.33i)T^{2} \)
23 \( 1 + (-1.02 - 8.23i)T + (-22.3 + 5.61i)T^{2} \)
29 \( 1 + (2.00 - 4.36i)T + (-18.8 - 22.0i)T^{2} \)
31 \( 1 + (-0.604 + 0.0186i)T + (30.9 - 1.90i)T^{2} \)
37 \( 1 + (-3.83 - 4.20i)T + (-3.41 + 36.8i)T^{2} \)
41 \( 1 + (1.14 + 0.106i)T + (40.3 + 7.53i)T^{2} \)
43 \( 1 + (2.95 + 9.27i)T + (-35.0 + 24.8i)T^{2} \)
47 \( 1 - 7.74T + 47T^{2} \)
53 \( 1 + (-6.13 - 9.25i)T + (-20.6 + 48.8i)T^{2} \)
59 \( 1 + (-0.324 + 0.838i)T + (-43.6 - 39.7i)T^{2} \)
61 \( 1 + (6.88 + 8.03i)T + (-9.35 + 60.2i)T^{2} \)
67 \( 1 + (-4.64 + 0.720i)T + (63.8 - 20.3i)T^{2} \)
71 \( 1 + (0.622 + 0.440i)T + (23.5 + 66.9i)T^{2} \)
73 \( 1 + (-5.35 - 0.496i)T + (71.7 + 13.4i)T^{2} \)
79 \( 1 + (0.871 - 0.616i)T + (26.2 - 74.5i)T^{2} \)
83 \( 1 + (8.31 + 1.29i)T + (79.0 + 25.1i)T^{2} \)
89 \( 1 + (-0.504 + 1.77i)T + (-75.6 - 46.8i)T^{2} \)
97 \( 1 + (1.69 - 1.98i)T + (-14.8 - 95.8i)T^{2} \)
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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

−9.812422305832691381852241291079, −9.351726680299918293924302943266, −8.932376829137899922048064748959, −7.977461624980346681924163105115, −7.23162191914095460804599546246, −5.86519800135475679807641839789, −5.32256269648093585079727926841, −3.60029130059820114184790151763, −3.21714067029104141199298885159, −1.20230240016385271734060182021, 0.846388732369676947580328878879, 2.02942395390057536290361023203, 2.94652848060389364028406274066, 4.27192780589636303257542900016, 5.84586966707414447111437038746, 6.73906399077377989613307874443, 7.33748156002209905753623715816, 8.325260097360710543277232208626, 9.350264979989417452196581008162, 9.652283141341667213542452573645

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