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Properties

Label 2-921-921.131-c1-0-63
Degree $2$
Conductor $921$
Sign $-0.994 + 0.101i$
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 + (−0.550 + 4.85i)7-s + (2.94 + 0.551i)9-s + (−0.844 + 3.35i)12-s + (−2.36 − 5.76i)13-s + (−3.81 − 1.21i)16-s + (−0.0536 − 0.244i)19-s + (1.72 − 8.28i)21-s + (−0.665 − 4.95i)25-s + (−4.99 − 1.42i)27-s + (9.42 + 2.57i)28-s + (−0.229 + 2.79i)31-s + (1.99 − 5.65i)36-s + (−11.3 − 3.35i)37-s + ⋯
L(s)  = 1  + (−0.995 − 0.0922i)3-s + (0.153 − 0.988i)4-s + (−0.208 + 1.83i)7-s + (0.982 + 0.183i)9-s + (−0.243 + 0.969i)12-s + (−0.657 − 1.59i)13-s + (−0.952 − 0.303i)16-s + (−0.0123 − 0.0562i)19-s + (0.376 − 1.80i)21-s + (−0.133 − 0.991i)25-s + (−0.961 − 0.273i)27-s + (1.78 + 0.486i)28-s + (−0.0413 + 0.501i)31-s + (0.332 − 0.943i)36-s + (−1.86 − 0.551i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.101i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(921\)    =    \(3 \cdot 307\)
Sign: $-0.994 + 0.101i$
Analytic conductor: \(7.35422\)
Root analytic conductor: \(2.71186\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{921} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 921,\ (\ :1/2),\ -0.994 + 0.101i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0117648 - 0.230795i\)
\(L(\frac12)\) \(\approx\) \(0.0117648 - 0.230795i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.665 + 4.95i)T^{2} \)
7 \( 1 + (0.550 - 4.85i)T + (-6.82 - 1.56i)T^{2} \)
11 \( 1 + (-9.46 + 5.59i)T^{2} \)
13 \( 1 + (2.36 + 5.76i)T + (-9.23 + 9.14i)T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.0536 + 0.244i)T + (-17.2 + 7.94i)T^{2} \)
23 \( 1 + (21.7 - 7.42i)T^{2} \)
29 \( 1 + (15.0 + 24.8i)T^{2} \)
31 \( 1 + (0.229 - 2.79i)T + (-30.5 - 5.06i)T^{2} \)
37 \( 1 + (11.3 + 3.35i)T + (31.0 + 20.1i)T^{2} \)
41 \( 1 + (-2.10 - 40.9i)T^{2} \)
43 \( 1 + (9.76 - 8.71i)T + (4.84 - 42.7i)T^{2} \)
47 \( 1 + (-26.7 - 38.6i)T^{2} \)
53 \( 1 + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (34.5 - 47.8i)T^{2} \)
61 \( 1 + (-0.0174 + 0.120i)T + (-58.4 - 17.2i)T^{2} \)
67 \( 1 + (5.09 + 10.4i)T + (-41.4 + 52.6i)T^{2} \)
71 \( 1 + (57.0 - 42.2i)T^{2} \)
73 \( 1 + (3.95 - 5.13i)T + (-18.5 - 70.6i)T^{2} \)
79 \( 1 + (-0.287 + 9.33i)T + (-78.8 - 4.86i)T^{2} \)
83 \( 1 + (-69.6 + 45.1i)T^{2} \)
89 \( 1 + (88.0 + 12.7i)T^{2} \)
97 \( 1 + (3.92 - 17.9i)T + (-88.1 - 40.5i)T^{2} \)
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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

−9.921079038469320347050118773015, −8.996049858604257795742620997431, −8.004546377749305296134049635860, −6.77085513954497826185329416335, −6.02016712852840401260493668098, −5.38078101702529861759957171439, −4.87466917032988705243524890776, −2.94912173813576257533132310788, −1.81233621930017561235241142139, −0.11950710591906776102898620563, 1.68265197609527264025684219863, 3.56703089669908991439845404721, 4.17986517733649230145534275591, 5.02734134777445918850306672377, 6.58943102834435213612139268570, 7.03957406625516372470400255625, 7.56685875725349313004060884588, 8.868044937145215664058011434412, 9.870364663163955532887863278675, 10.51565147145241287078534933699

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