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Properties

Label 2-920-115.102-c1-0-3
Degree $2$
Conductor $920$
Sign $-0.914 - 0.404i$
Analytic cond. $7.34623$
Root an. cond. $2.71039$
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  + (2.32 − 0.505i)3-s + (−2.23 − 0.101i)5-s + (−2.56 + 1.40i)7-s + (2.42 − 1.10i)9-s + (−4.43 − 3.83i)11-s + (−2.71 + 4.97i)13-s + (−5.24 + 0.895i)15-s + (−4.66 − 3.49i)17-s + (−0.842 + 5.85i)19-s + (−5.26 + 4.55i)21-s + (1.25 + 4.62i)23-s + (4.97 + 0.451i)25-s + (−0.640 + 0.479i)27-s + (3.20 − 0.461i)29-s + (3.85 − 2.47i)31-s + ⋯
L(s)  = 1  + (1.34 − 0.292i)3-s + (−0.998 − 0.0451i)5-s + (−0.970 + 0.529i)7-s + (0.807 − 0.368i)9-s + (−1.33 − 1.15i)11-s + (−0.752 + 1.37i)13-s + (−1.35 + 0.231i)15-s + (−1.13 − 0.847i)17-s + (−0.193 + 1.34i)19-s + (−1.14 + 0.994i)21-s + (0.260 + 0.965i)23-s + (0.995 + 0.0902i)25-s + (−0.123 + 0.0922i)27-s + (0.595 − 0.0856i)29-s + (0.691 − 0.444i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.914 - 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $-0.914 - 0.404i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ -0.914 - 0.404i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0512849 + 0.242760i\)
\(L(\frac12)\) \(\approx\) \(0.0512849 + 0.242760i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.23 + 0.101i)T \)
23 \( 1 + (-1.25 - 4.62i)T \)
good3 \( 1 + (-2.32 + 0.505i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (2.56 - 1.40i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (4.43 + 3.83i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (2.71 - 4.97i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (4.66 + 3.49i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (0.842 - 5.85i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-3.20 + 0.461i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (-3.85 + 2.47i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (3.23 + 1.20i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (-2.20 + 4.81i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (0.300 + 1.38i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (8.32 + 8.32i)T + 47iT^{2} \)
53 \( 1 + (-4.08 - 7.48i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (1.62 + 5.54i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (1.49 + 2.32i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (-9.03 - 0.646i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (5.72 + 6.61i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (4.01 + 5.36i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (6.67 - 1.95i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-1.58 + 4.24i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (-2.65 - 1.70i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (1.50 + 4.03i)T + (-73.3 + 63.5i)T^{2} \)
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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

−10.28109277940909878576855391360, −9.277491601816122454029459452377, −8.770483738434991866277917752212, −7.991041506620066803063641636983, −7.31658546586920644148114633265, −6.37478031926915445330344811039, −5.08067770870389160968395911360, −3.83540864541382699033132378522, −3.04071620879013389383167152767, −2.24759005045108488226850477954, 0.089370786682569735603506839940, 2.66119498340901345011987780264, 2.96706666600558237600513098528, 4.25922225545308313607913272917, 4.88438615723723769929637013054, 6.66107213067117193154240512384, 7.34034913820038700750331543921, 8.159121459189070086309008880195, 8.665883703682216439634999885690, 9.861645435958401655542089920957

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