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Properties

Label 2-921-921.92-c1-0-45
Degree $2$
Conductor $921$
Sign $0.696 - 0.717i$
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  + (−0.625 + 1.61i)3-s + (1.81 − 0.835i)4-s + (0.914 + 2.04i)7-s + (−2.21 − 2.02i)9-s + (0.213 + 3.45i)12-s + (5.24 − 4.31i)13-s + (2.60 − 3.03i)16-s + (1.88 + 5.33i)19-s + (−3.86 + 0.198i)21-s + (1.75 + 4.68i)25-s + (4.65 − 2.31i)27-s + (3.36 + 2.94i)28-s + (0.0122 + 0.595i)31-s + (−5.71 − 1.81i)36-s + (−0.429 − 5.96i)37-s + ⋯
L(s)  = 1  + (−0.361 + 0.932i)3-s + (0.908 − 0.417i)4-s + (0.345 + 0.771i)7-s + (−0.739 − 0.673i)9-s + (0.0615 + 0.998i)12-s + (1.45 − 1.19i)13-s + (0.650 − 0.759i)16-s + (0.431 + 1.22i)19-s + (−0.844 + 0.0433i)21-s + (0.351 + 0.936i)25-s + (0.895 − 0.445i)27-s + (0.636 + 0.556i)28-s + (0.00219 + 0.106i)31-s + (−0.952 − 0.303i)36-s + (−0.0705 − 0.979i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(921\)    =    \(3 \cdot 307\)
Sign: $0.696 - 0.717i$
Analytic conductor: \(7.35422\)
Root analytic conductor: \(2.71186\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{921} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 921,\ (\ :1/2),\ 0.696 - 0.717i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74064 + 0.736044i\)
\(L(\frac12)\) \(\approx\) \(1.74064 + 0.736044i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.625 - 1.61i)T \)
307 \( 1 + (16.6 + 5.51i)T \)
good2 \( 1 + (-1.81 + 0.835i)T^{2} \)
5 \( 1 + (-1.75 - 4.68i)T^{2} \)
7 \( 1 + (-0.914 - 2.04i)T + (-4.66 + 5.22i)T^{2} \)
11 \( 1 + (-1.46 + 10.9i)T^{2} \)
13 \( 1 + (-5.24 + 4.31i)T + (2.51 - 12.7i)T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.88 - 5.33i)T + (-14.8 + 11.9i)T^{2} \)
23 \( 1 + (22.9 + 1.88i)T^{2} \)
29 \( 1 + (7.36 + 28.0i)T^{2} \)
31 \( 1 + (-0.0122 - 0.595i)T + (-30.9 + 1.27i)T^{2} \)
37 \( 1 + (0.429 + 5.96i)T + (-36.6 + 5.29i)T^{2} \)
41 \( 1 + (-15.2 - 38.0i)T^{2} \)
43 \( 1 + (6.79 - 10.4i)T + (-17.5 - 39.2i)T^{2} \)
47 \( 1 + (24.3 - 40.2i)T^{2} \)
53 \( 1 + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (-13.8 + 57.3i)T^{2} \)
61 \( 1 + (-10.7 - 10.0i)T + (4.38 + 60.8i)T^{2} \)
67 \( 1 + (11.0 + 1.25i)T + (65.2 + 15.0i)T^{2} \)
71 \( 1 + (-41.6 - 57.5i)T^{2} \)
73 \( 1 + (2.75 + 16.6i)T + (-69.0 + 23.5i)T^{2} \)
79 \( 1 + (-6.92 - 16.3i)T + (-54.9 + 56.7i)T^{2} \)
83 \( 1 + (82.1 + 11.8i)T^{2} \)
89 \( 1 + (-60.6 + 65.1i)T^{2} \)
97 \( 1 + (5.76 - 16.3i)T + (-75.5 - 60.8i)T^{2} \)
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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

−10.38571778376920113057180009060, −9.506858228309603042957649678162, −8.566583398486201800789532225414, −7.78007409272143866068368507632, −6.44710795997201705839300399480, −5.68956510004801772965065821485, −5.27739392410264798468773097384, −3.74154988720318107241408366920, −2.90107388803520595216874755442, −1.34080009865332449318343695559, 1.13145278261320823353355131976, 2.19348373323430876220848989401, 3.46044162481040689494043409128, 4.65381964615919889652522267516, 5.97275999902197822451834181403, 6.78210216078395632932140605776, 7.14520557760972640955785752319, 8.205403326081060949993373852788, 8.802384972835115771762147522618, 10.29007011705823293695120718056

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