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Properties

Label 2-920-115.102-c1-0-32
Degree $2$
Conductor $920$
Sign $0.583 + 0.812i$
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  + (3.15 − 0.687i)3-s + (−0.353 − 2.20i)5-s + (−2.17 + 1.18i)7-s + (6.77 − 3.09i)9-s + (3.54 + 3.06i)11-s + (3.30 − 6.06i)13-s + (−2.63 − 6.73i)15-s + (−2.11 − 1.58i)17-s + (−0.781 + 5.43i)19-s + (−6.05 + 5.24i)21-s + (−4.27 − 2.17i)23-s + (−4.74 + 1.56i)25-s + (11.5 − 8.61i)27-s + (2.86 − 0.411i)29-s + (−5.61 + 3.60i)31-s + ⋯
L(s)  = 1  + (1.82 − 0.396i)3-s + (−0.158 − 0.987i)5-s + (−0.822 + 0.449i)7-s + (2.25 − 1.03i)9-s + (1.06 + 0.924i)11-s + (0.917 − 1.68i)13-s + (−0.680 − 1.73i)15-s + (−0.512 − 0.383i)17-s + (−0.179 + 1.24i)19-s + (−1.32 + 1.14i)21-s + (−0.890 − 0.454i)23-s + (−0.949 + 0.312i)25-s + (2.21 − 1.65i)27-s + (0.531 − 0.0764i)29-s + (−1.00 + 0.647i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.583 + 0.812i$
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: \((2,\ 920,\ (\ :1/2),\ 0.583 + 0.812i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.51844 - 1.29158i\)
\(L(\frac12)\) \(\approx\) \(2.51844 - 1.29158i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.353 + 2.20i)T \)
23 \( 1 + (4.27 + 2.17i)T \)
good3 \( 1 + (-3.15 + 0.687i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (2.17 - 1.18i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (-3.54 - 3.06i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (-3.30 + 6.06i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (2.11 + 1.58i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (0.781 - 5.43i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-2.86 + 0.411i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (5.61 - 3.60i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-4.79 - 1.78i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (-1.93 + 4.23i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (1.78 + 8.20i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (0.664 + 0.664i)T + 47iT^{2} \)
53 \( 1 + (-2.19 - 4.02i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (-2.47 - 8.42i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (-1.51 - 2.35i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (-4.30 - 0.307i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (-6.78 - 7.83i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (2.67 + 3.56i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (-0.0960 + 0.0281i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (3.10 - 8.31i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (5.99 + 3.85i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-1.63 - 4.37i)T + (-73.3 + 63.5i)T^{2} \)
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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

−9.663684804697667323765432920209, −8.947822839041969110209907306407, −8.418048578782330742098376097561, −7.72403313960908971792929464902, −6.73352119163331275732952604614, −5.68140204932771540709122765595, −4.14019866879817246597185548361, −3.58408925851563512336398203476, −2.40744728192459445783267449123, −1.26489441926594558728580261247, 1.85275057143932113244124531316, 2.97600364632085030682558143524, 3.83097246597823012077313429660, 4.17010696251551515790890237056, 6.44718324418621793384129396478, 6.71146431053883224990430674513, 7.82240627237867060680129889280, 8.696269331763313790650585705453, 9.351583106161865145892136022756, 9.835926468560406947361714697250

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