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Properties

Label 2-921-921.47-c1-0-54
Degree $2$
Conductor $921$
Sign $-0.161 + 0.986i$
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  + (−1.72 + 0.159i)3-s + (0.306 + 1.97i)4-s + (−3.92 + 2.90i)7-s + (2.94 − 0.551i)9-s + (−0.844 − 3.35i)12-s + (−3.80 − 4.93i)13-s + (−3.81 + 1.21i)16-s + (−1.58 + 7.24i)19-s + (6.30 − 5.63i)21-s + (4.62 − 1.90i)25-s + (−4.99 + 1.42i)27-s + (−6.94 − 6.87i)28-s + (8.82 − 6.11i)31-s + (1.99 + 5.65i)36-s + (2.76 − 11.4i)37-s + ⋯
L(s)  = 1  + (−0.995 + 0.0922i)3-s + (0.153 + 0.988i)4-s + (−1.48 + 1.09i)7-s + (0.982 − 0.183i)9-s + (−0.243 − 0.969i)12-s + (−1.05 − 1.36i)13-s + (−0.952 + 0.303i)16-s + (−0.364 + 1.66i)19-s + (1.37 − 1.22i)21-s + (0.924 − 0.380i)25-s + (−0.961 + 0.273i)27-s + (−1.31 − 1.29i)28-s + (1.58 − 1.09i)31-s + (0.332 + 0.943i)36-s + (0.454 − 1.89i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(921\)    =    \(3 \cdot 307\)
Sign: $-0.161 + 0.986i$
Analytic conductor: \(7.35422\)
Root analytic conductor: \(2.71186\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{921} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 921,\ (\ :1/2),\ -0.161 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0711209 - 0.0837347i\)
\(L(\frac12)\) \(\approx\) \(0.0711209 - 0.0837347i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.72 - 0.159i)T \)
307 \( 1 + (2.47 + 17.3i)T \)
good2 \( 1 + (-0.306 - 1.97i)T^{2} \)
5 \( 1 + (-4.62 + 1.90i)T^{2} \)
7 \( 1 + (3.92 - 2.90i)T + (2.05 - 6.69i)T^{2} \)
11 \( 1 + (-0.112 + 10.9i)T^{2} \)
13 \( 1 + (3.80 + 4.93i)T + (-3.30 + 12.5i)T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.58 - 7.24i)T + (-17.2 - 7.94i)T^{2} \)
23 \( 1 + (-17.3 + 15.1i)T^{2} \)
29 \( 1 + (13.9 + 25.4i)T^{2} \)
31 \( 1 + (-8.82 + 6.11i)T + (10.9 - 29.0i)T^{2} \)
37 \( 1 + (-2.76 + 11.4i)T + (-32.9 - 16.8i)T^{2} \)
41 \( 1 + (-34.4 - 22.2i)T^{2} \)
43 \( 1 + (5.54 - 1.82i)T + (34.5 - 25.5i)T^{2} \)
47 \( 1 + (-20.0 - 42.4i)T^{2} \)
53 \( 1 + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (-58.6 + 6.04i)T^{2} \)
61 \( 1 + (10.6 - 8.40i)T + (14.2 - 59.3i)T^{2} \)
67 \( 1 + (2.30 + 3.40i)T + (-24.8 + 62.2i)T^{2} \)
71 \( 1 + (-65.0 + 28.3i)T^{2} \)
73 \( 1 + (16.7 - 2.25i)T + (70.4 - 19.2i)T^{2} \)
79 \( 1 + (0.547 + 17.7i)T + (-78.8 + 4.86i)T^{2} \)
83 \( 1 + (73.9 - 37.7i)T^{2} \)
89 \( 1 + (-55.0 - 69.9i)T^{2} \)
97 \( 1 + (-0.646 - 2.95i)T + (-88.1 + 40.5i)T^{2} \)
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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

−9.991841974506014357396141526095, −9.115614975794298925129032967008, −8.059601691386998977927607334273, −7.25676518113047049477489680063, −6.20046900692687801132416794201, −5.76843888486651637946071576549, −4.51066384503833842940523289051, −3.34711573042201282425214015334, −2.48438593466370379317870043716, −0.06395239789869995443114595039, 1.18562823427998931620172512216, 2.83537756847059117216050564120, 4.48797852822864698363271511943, 4.88739997761088067988360574547, 6.36042782754126266349762817707, 6.72315163448588784953733519249, 7.17723532616879128733550192603, 8.994551716912979619260181341764, 9.839505036249953954128281682841, 10.18009441547383584806536933600

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