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Properties

Label 2-920-115.102-c1-0-12
Degree $2$
Conductor $920$
Sign $0.0382 - 0.999i$
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  + (0.595 − 0.129i)3-s + (0.588 + 2.15i)5-s + (1.06 − 0.580i)7-s + (−2.39 + 1.09i)9-s + (−1.50 − 1.30i)11-s + (−1.77 + 3.25i)13-s + (0.629 + 1.20i)15-s + (4.73 + 3.54i)17-s + (−0.292 + 2.03i)19-s + (0.558 − 0.483i)21-s + (1.55 + 4.53i)23-s + (−4.30 + 2.53i)25-s + (−2.74 + 2.05i)27-s + (2.67 − 0.385i)29-s + (4.70 − 3.02i)31-s + ⋯
L(s)  = 1  + (0.343 − 0.0747i)3-s + (0.263 + 0.964i)5-s + (0.402 − 0.219i)7-s + (−0.797 + 0.363i)9-s + (−0.453 − 0.392i)11-s + (−0.493 + 0.903i)13-s + (0.162 + 0.311i)15-s + (1.14 + 0.860i)17-s + (−0.0669 + 0.465i)19-s + (0.121 − 0.105i)21-s + (0.324 + 0.945i)23-s + (−0.861 + 0.507i)25-s + (−0.528 + 0.395i)27-s + (0.497 − 0.0715i)29-s + (0.844 − 0.542i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0382 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0382 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.0382 - 0.999i$
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: \((2,\ 920,\ (\ :1/2),\ 0.0382 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12626 + 1.08400i\)
\(L(\frac12)\) \(\approx\) \(1.12626 + 1.08400i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.588 - 2.15i)T \)
23 \( 1 + (-1.55 - 4.53i)T \)
good3 \( 1 + (-0.595 + 0.129i)T + (2.72 - 1.24i)T^{2} \)
7 \( 1 + (-1.06 + 0.580i)T + (3.78 - 5.88i)T^{2} \)
11 \( 1 + (1.50 + 1.30i)T + (1.56 + 10.8i)T^{2} \)
13 \( 1 + (1.77 - 3.25i)T + (-7.02 - 10.9i)T^{2} \)
17 \( 1 + (-4.73 - 3.54i)T + (4.78 + 16.3i)T^{2} \)
19 \( 1 + (0.292 - 2.03i)T + (-18.2 - 5.35i)T^{2} \)
29 \( 1 + (-2.67 + 0.385i)T + (27.8 - 8.17i)T^{2} \)
31 \( 1 + (-4.70 + 3.02i)T + (12.8 - 28.1i)T^{2} \)
37 \( 1 + (5.47 + 2.04i)T + (27.9 + 24.2i)T^{2} \)
41 \( 1 + (1.19 - 2.61i)T + (-26.8 - 30.9i)T^{2} \)
43 \( 1 + (-1.08 - 5.00i)T + (-39.1 + 17.8i)T^{2} \)
47 \( 1 + (2.60 + 2.60i)T + 47iT^{2} \)
53 \( 1 + (6.56 + 12.0i)T + (-28.6 + 44.5i)T^{2} \)
59 \( 1 + (0.547 + 1.86i)T + (-49.6 + 31.8i)T^{2} \)
61 \( 1 + (-4.84 - 7.53i)T + (-25.3 + 55.4i)T^{2} \)
67 \( 1 + (-14.2 - 1.01i)T + (66.3 + 9.53i)T^{2} \)
71 \( 1 + (-8.83 - 10.1i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-3.02 - 4.04i)T + (-20.5 + 70.0i)T^{2} \)
79 \( 1 + (12.2 - 3.60i)T + (66.4 - 42.7i)T^{2} \)
83 \( 1 + (-1.98 + 5.31i)T + (-62.7 - 54.3i)T^{2} \)
89 \( 1 + (-3.99 - 2.56i)T + (36.9 + 80.9i)T^{2} \)
97 \( 1 + (-1.07 - 2.88i)T + (-73.3 + 63.5i)T^{2} \)
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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

−10.18242247652859511005271632386, −9.652994823814417504323415811915, −8.367770356837864869070196683602, −7.892468839893098453765690380862, −6.93991907548436377615147024642, −5.97537264382798195246226246443, −5.15746072472629315979772230812, −3.75828022203764556915361341055, −2.86912002913938172413270621413, −1.77783098118741403096797161529, 0.70233499766711380685635201995, 2.35585422885421779060464251746, 3.30807247155531243681163778870, 4.91423529695656531947806949672, 5.17389831603685350047967415161, 6.33895240778228175104356741443, 7.60888748259691200109913238756, 8.296560070707823317833872933427, 8.958072125512198312854665173367, 9.798405987751135846502813573749

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