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Properties

Label 2-920-115.102-c1-0-29
Degree $2$
Conductor $920$
Sign $0.976 + 0.215i$
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  + (2.44 − 0.531i)3-s + (1.93 + 1.11i)5-s + (0.950 − 0.518i)7-s + (2.96 − 1.35i)9-s + (−0.259 − 0.225i)11-s + (1.11 − 2.05i)13-s + (5.32 + 1.70i)15-s + (−3.79 − 2.83i)17-s + (0.327 − 2.27i)19-s + (2.04 − 1.77i)21-s + (3.87 + 2.82i)23-s + (2.49 + 4.33i)25-s + (0.512 − 0.383i)27-s + (−5.37 + 0.772i)29-s + (0.720 − 0.462i)31-s + ⋯
L(s)  = 1  + (1.41 − 0.306i)3-s + (0.865 + 0.500i)5-s + (0.359 − 0.196i)7-s + (0.987 − 0.450i)9-s + (−0.0783 − 0.0679i)11-s + (0.310 − 0.568i)13-s + (1.37 + 0.440i)15-s + (−0.920 − 0.688i)17-s + (0.0751 − 0.522i)19-s + (0.446 − 0.386i)21-s + (0.808 + 0.588i)23-s + (0.498 + 0.866i)25-s + (0.0985 − 0.0737i)27-s + (−0.998 + 0.143i)29-s + (0.129 − 0.0831i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.976 + 0.215i$
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: \((2,\ 920,\ (\ :1/2),\ 0.976 + 0.215i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.95375 - 0.322334i\)
\(L(\frac12)\) \(\approx\) \(2.95375 - 0.322334i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.93 - 1.11i)T \)
23 \( 1 + (-3.87 - 2.82i)T \)
good3 \( 1 + (-2.44 + 0.531i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (-0.950 + 0.518i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (0.259 + 0.225i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (-1.11 + 2.05i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (3.79 + 2.83i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (-0.327 + 2.27i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (5.37 - 0.772i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (-0.720 + 0.462i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-2.52 - 0.940i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (0.890 - 1.95i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-1.89 - 8.69i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (2.23 + 2.23i)T + 47iT^{2} \)
53 \( 1 + (3.06 + 5.60i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (-1.91 - 6.51i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (-1.27 - 1.98i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (-2.92 - 0.209i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (4.80 + 5.54i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (6.29 + 8.40i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (9.49 - 2.78i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (1.76 - 4.73i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (-4.22 - 2.71i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (0.330 + 0.884i)T + (-73.3 + 63.5i)T^{2} \)
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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

−9.738170149918231417441880786052, −9.233752037477065856249022313482, −8.438464074756018927256404056313, −7.53956922694445969911549683861, −6.90007007331321753064406721748, −5.78390698417519100028929550160, −4.65805375766706480391724588339, −3.27701899448857535399679031892, −2.62935970963532101184149658870, −1.53278581001970955257843800417, 1.70447743866520745772327005375, 2.46902839792852057824932565382, 3.74700043674944949867619968793, 4.62407306222720017047539684428, 5.72151980368023160899659198412, 6.74912018847213204956077481576, 7.88811636331790394582097825368, 8.746286020875397393375106779609, 9.006981135630975484228525038545, 9.885048780743356704523468984588

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