Properties

Label 2-920-115.102-c1-0-15
Degree $2$
Conductor $920$
Sign $0.469 - 0.882i$
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  + (2.31 − 0.503i)3-s + (−2.17 + 0.497i)5-s + (−2.82 + 1.54i)7-s + (2.36 − 1.08i)9-s + (3.68 + 3.19i)11-s + (−0.318 + 0.583i)13-s + (−4.79 + 2.24i)15-s + (3.82 + 2.86i)17-s + (−0.133 + 0.925i)19-s + (−5.76 + 4.99i)21-s + (4.78 + 0.374i)23-s + (4.50 − 2.17i)25-s + (−0.751 + 0.562i)27-s + (−0.0387 + 0.00557i)29-s + (−5.58 + 3.58i)31-s + ⋯
L(s)  = 1  + (1.33 − 0.290i)3-s + (−0.974 + 0.222i)5-s + (−1.06 + 0.583i)7-s + (0.789 − 0.360i)9-s + (1.11 + 0.962i)11-s + (−0.0884 + 0.161i)13-s + (−1.23 + 0.580i)15-s + (0.926 + 0.693i)17-s + (−0.0305 + 0.212i)19-s + (−1.25 + 1.08i)21-s + (0.996 + 0.0780i)23-s + (0.900 − 0.434i)25-s + (−0.144 + 0.108i)27-s + (−0.00719 + 0.00103i)29-s + (−1.00 + 0.644i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.469 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.469 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.469 - 0.882i$
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: \((2,\ 920,\ (\ :1/2),\ 0.469 - 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55584 + 0.934704i\)
\(L(\frac12)\) \(\approx\) \(1.55584 + 0.934704i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.17 - 0.497i)T \)
23 \( 1 + (-4.78 - 0.374i)T \)
good3 \( 1 + (-2.31 + 0.503i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (2.82 - 1.54i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (-3.68 - 3.19i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (0.318 - 0.583i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (-3.82 - 2.86i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (0.133 - 0.925i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (0.0387 - 0.00557i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (5.58 - 3.58i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (-4.12 - 1.53i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (4.42 - 9.68i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-1.14 - 5.25i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (-0.848 - 0.848i)T + 47iT^{2} \)
53 \( 1 + (2.56 + 4.70i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (3.03 + 10.3i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (-2.62 - 4.09i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (-9.32 - 0.666i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (7.26 + 8.38i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (10.2 + 13.6i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (-16.1 + 4.74i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-2.36 + 6.34i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (5.83 + 3.74i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (1.78 + 4.79i)T + (-73.3 + 63.5i)T^{2} \)
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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

−9.835449534294497604955619178864, −9.332957410564980393163976760690, −8.561244247076809146428475872249, −7.75303237789685123870204981497, −7.01221136307964783273349565133, −6.21917473866292877851743513336, −4.64487831106698271974681290589, −3.49029436533992690254695844097, −3.09403397783113139988950032980, −1.67514425364128181592593880200, 0.77111830727422499254612188900, 2.82023913371800621248531416763, 3.59936793454162428828715462218, 4.01942446369101523918839618363, 5.52325118387470099882705243628, 6.86078429795024234978421778245, 7.45887015869783596731282627965, 8.419290565701854343140368110935, 9.095406003593802425572209761588, 9.601760672224501549976714662168

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