Properties

Label 2-927-927.77-c1-0-14
Degree $2$
Conductor $927$
Sign $-0.860 + 0.508i$
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  + (−1.01 + 0.430i)2-s + (1.10 + 1.33i)3-s + (−0.542 + 0.559i)4-s + (−1.85 + 1.31i)5-s + (−1.69 − 0.879i)6-s + (3.16 − 1.22i)7-s + (1.10 − 2.86i)8-s + (−0.552 + 2.94i)9-s + (1.31 − 2.13i)10-s + (−1.60 + 2.12i)11-s + (−1.34 − 0.104i)12-s + (−0.532 + 3.43i)13-s + (−2.69 + 2.60i)14-s + (−3.79 − 1.01i)15-s + (0.0564 + 1.83i)16-s + (−5.41 − 1.90i)17-s + ⋯
L(s)  = 1  + (−0.719 + 0.304i)2-s + (0.638 + 0.769i)3-s + (−0.271 + 0.279i)4-s + (−0.828 + 0.586i)5-s + (−0.693 − 0.359i)6-s + (1.19 − 0.463i)7-s + (0.392 − 1.01i)8-s + (−0.184 + 0.982i)9-s + (0.417 − 0.674i)10-s + (−0.483 + 0.640i)11-s + (−0.388 − 0.0300i)12-s + (−0.147 + 0.952i)13-s + (−0.718 + 0.697i)14-s + (−0.980 − 0.262i)15-s + (0.0141 + 0.457i)16-s + (−1.31 − 0.463i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 + 0.508i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.860 + 0.508i$
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: \((2,\ 927,\ (\ :1/2),\ -0.860 + 0.508i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.160133 - 0.585405i\)
\(L(\frac12)\) \(\approx\) \(0.160133 - 0.585405i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.10 - 1.33i)T \)
103 \( 1 + (-5.77 - 8.34i)T \)
good2 \( 1 + (1.01 - 0.430i)T + (1.39 - 1.43i)T^{2} \)
5 \( 1 + (1.85 - 1.31i)T + (1.66 - 4.71i)T^{2} \)
7 \( 1 + (-3.16 + 1.22i)T + (5.17 - 4.71i)T^{2} \)
11 \( 1 + (1.60 - 2.12i)T + (-3.01 - 10.5i)T^{2} \)
13 \( 1 + (0.532 - 3.43i)T + (-12.3 - 3.94i)T^{2} \)
17 \( 1 + (5.41 + 1.90i)T + (13.2 + 10.6i)T^{2} \)
19 \( 1 + (0.452 - 0.0279i)T + (18.8 - 2.33i)T^{2} \)
23 \( 1 + (0.0661 + 0.534i)T + (-22.3 + 5.61i)T^{2} \)
29 \( 1 + (-0.0946 + 0.205i)T + (-18.8 - 22.0i)T^{2} \)
31 \( 1 + (2.86 - 0.0883i)T + (30.9 - 1.90i)T^{2} \)
37 \( 1 + (-7.71 - 8.46i)T + (-3.41 + 36.8i)T^{2} \)
41 \( 1 + (3.83 + 0.355i)T + (40.3 + 7.53i)T^{2} \)
43 \( 1 + (3.11 + 9.80i)T + (-35.0 + 24.8i)T^{2} \)
47 \( 1 + 2.28T + 47T^{2} \)
53 \( 1 + (0.324 + 0.489i)T + (-20.6 + 48.8i)T^{2} \)
59 \( 1 + (-0.361 + 0.933i)T + (-43.6 - 39.7i)T^{2} \)
61 \( 1 + (6.69 + 7.81i)T + (-9.35 + 60.2i)T^{2} \)
67 \( 1 + (4.71 - 0.731i)T + (63.8 - 20.3i)T^{2} \)
71 \( 1 + (2.24 + 1.59i)T + (23.5 + 66.9i)T^{2} \)
73 \( 1 + (-7.64 - 0.708i)T + (71.7 + 13.4i)T^{2} \)
79 \( 1 + (7.61 - 5.39i)T + (26.2 - 74.5i)T^{2} \)
83 \( 1 + (-6.14 - 0.954i)T + (79.0 + 25.1i)T^{2} \)
89 \( 1 + (0.236 - 0.831i)T + (-75.6 - 46.8i)T^{2} \)
97 \( 1 + (-11.3 + 13.2i)T + (-14.8 - 95.8i)T^{2} \)
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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

−10.42514058277758333928241204860, −9.577975975875713896797737308271, −8.791721961967377999081223455570, −8.089864028563930693121957508797, −7.47187007477688405190625163852, −6.82684866335811008661285098464, −4.77362159063820228872937141367, −4.44643539145951991793080145622, −3.46060778270533843410979536682, −2.03911364039195881457351765024, 0.34064732162876386165423602338, 1.58971228355249117575699116021, 2.66130577723726741313154799082, 4.19120924115437059107015731748, 5.13217699296835032557474557789, 6.11422716667198886949314346082, 7.72180575998867127478717009965, 7.995081140744841389705497864731, 8.673970669138092084227902706484, 9.188831555929024893354288409565

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