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Properties

Label 2-921-921.116-c1-0-11
Degree $2$
Conductor $921$
Sign $-0.848 - 0.529i$
Analytic cond. $7.35422$
Root an. cond. $2.71186$
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  + (−1.55 + 0.772i)3-s + (−1.93 + 0.487i)4-s + (0.492 + 0.380i)7-s + (1.80 − 2.39i)9-s + (2.63 − 2.25i)12-s + (4.49 + 4.01i)13-s + (3.52 − 1.89i)16-s + (−0.0253 + 0.821i)19-s + (−1.05 − 0.208i)21-s + (−4.89 + 1.01i)25-s + (−0.954 + 5.10i)27-s + (−1.14 − 0.496i)28-s + (−5.62 + 1.35i)31-s + (−2.33 + 5.52i)36-s + (−3.92 + 0.161i)37-s + ⋯
L(s)  = 1  + (−0.895 + 0.445i)3-s + (−0.969 + 0.243i)4-s + (0.186 + 0.143i)7-s + (0.602 − 0.798i)9-s + (0.759 − 0.650i)12-s + (1.24 + 1.11i)13-s + (0.881 − 0.473i)16-s + (−0.00580 + 0.188i)19-s + (−0.230 − 0.0455i)21-s + (−0.978 + 0.203i)25-s + (−0.183 + 0.982i)27-s + (−0.215 − 0.0938i)28-s + (−1.01 + 0.243i)31-s + (−0.389 + 0.920i)36-s + (−0.645 + 0.0265i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(921\)    =    \(3 \cdot 307\)
Sign: $-0.848 - 0.529i$
Analytic conductor: \(7.35422\)
Root analytic conductor: \(2.71186\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{921} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 921,\ (\ :1/2),\ -0.848 - 0.529i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.151277 + 0.527942i\)
\(L(\frac12)\) \(\approx\) \(0.151277 + 0.527942i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.55 - 0.772i)T \)
307 \( 1 + (8.57 + 15.2i)T \)
good2 \( 1 + (1.93 - 0.487i)T^{2} \)
5 \( 1 + (4.89 - 1.01i)T^{2} \)
7 \( 1 + (-0.492 - 0.380i)T + (1.77 + 6.77i)T^{2} \)
11 \( 1 + (-7.49 + 8.05i)T^{2} \)
13 \( 1 + (-4.49 - 4.01i)T + (1.46 + 12.9i)T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0253 - 0.821i)T + (-18.9 - 1.16i)T^{2} \)
23 \( 1 + (13.4 + 18.6i)T^{2} \)
29 \( 1 + (-24.0 + 16.2i)T^{2} \)
31 \( 1 + (5.62 - 1.35i)T + (27.6 - 14.1i)T^{2} \)
37 \( 1 + (3.92 - 0.161i)T + (36.8 - 3.03i)T^{2} \)
41 \( 1 + (17.5 + 37.0i)T^{2} \)
43 \( 1 + (3.60 - 10.5i)T + (-34.0 - 26.2i)T^{2} \)
47 \( 1 + (-34.0 - 32.3i)T^{2} \)
53 \( 1 + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-20.7 + 55.2i)T^{2} \)
61 \( 1 + (-0.0172 - 0.839i)T + (-60.9 + 2.50i)T^{2} \)
67 \( 1 + (0.558 - 0.990i)T + (-34.6 - 57.3i)T^{2} \)
71 \( 1 + (-9.44 - 70.3i)T^{2} \)
73 \( 1 + (2.30 - 0.759i)T + (58.7 - 43.3i)T^{2} \)
79 \( 1 + (13.0 - 10.5i)T + (16.9 - 77.1i)T^{2} \)
83 \( 1 + (-82.7 - 6.80i)T^{2} \)
89 \( 1 + (88.9 - 1.82i)T^{2} \)
97 \( 1 + (-0.495 - 16.0i)T + (-96.8 + 5.97i)T^{2} \)
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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

−10.38932669167982643535011840949, −9.534230809109180352534155790365, −8.938204171881906612665660302578, −8.058093944012590980767663555879, −6.87503869883163353658065571169, −5.95035652458047728702475390833, −5.14451027606051814563354316541, −4.19372023435905405069723703326, −3.56021396718530420037408599672, −1.44233106931777658940734194209, 0.33398847601676890109862627409, 1.60361697090562089731110298643, 3.51757541748898568988777743338, 4.51052621974809112907544785339, 5.55226818243571740001542865925, 5.95302725131974515022265627855, 7.20175039970985615067742704560, 8.084583751671848589744864886100, 8.806952475791026105504115993271, 9.927658342001185084960405012373

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