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Properties

Label 2-456-57.50-c1-0-15
Degree $2$
Conductor $456$
Sign $0.975 + 0.220i$
Analytic cond. $3.64117$
Root an. cond. $1.90818$
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  + (1.73 + 0.0649i)3-s + (3.15 − 1.82i)5-s + 1.32·7-s + (2.99 + 0.224i)9-s + 5.22i·11-s + (−5.77 − 3.33i)13-s + (5.58 − 2.95i)15-s + (−4.40 + 2.54i)17-s + (−3.57 − 2.49i)19-s + (2.29 + 0.0862i)21-s + (−2.14 − 1.23i)23-s + (4.14 − 7.18i)25-s + (5.16 + 0.583i)27-s + (−0.559 + 0.969i)29-s + 0.304i·31-s + ⋯
L(s)  = 1  + (0.999 + 0.0375i)3-s + (1.41 − 0.815i)5-s + 0.501·7-s + (0.997 + 0.0749i)9-s + 1.57i·11-s + (−1.60 − 0.925i)13-s + (1.44 − 0.761i)15-s + (−1.06 + 0.616i)17-s + (−0.819 − 0.573i)19-s + (0.501 + 0.0188i)21-s + (−0.447 − 0.258i)23-s + (0.829 − 1.43i)25-s + (0.993 + 0.112i)27-s + (−0.103 + 0.179i)29-s + 0.0546i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.975 + 0.220i$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ 0.975 + 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32318 - 0.259607i\)
\(L(\frac12)\) \(\approx\) \(2.32318 - 0.259607i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.73 - 0.0649i)T \)
19 \( 1 + (3.57 + 2.49i)T \)
good5 \( 1 + (-3.15 + 1.82i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 1.32T + 7T^{2} \)
11 \( 1 - 5.22iT - 11T^{2} \)
13 \( 1 + (5.77 + 3.33i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.40 - 2.54i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.14 + 1.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.559 - 0.969i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.304iT - 31T^{2} \)
37 \( 1 - 4.66iT - 37T^{2} \)
41 \( 1 + (2.16 + 3.74i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.93 - 8.54i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-7.04 - 4.06i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.391 - 0.677i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.58 + 4.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.21 + 12.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.10 - 1.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (2.10 + 3.65i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.88 + 8.45i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.31 + 2.48i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.31iT - 83T^{2} \)
89 \( 1 + (0.227 - 0.393i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.45 + 5.45i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.66091098108496697492196726526, −9.856919973054196056594965927124, −9.386973783099126023044260746411, −8.438892499081786226707078491226, −7.51884925407330743341895200590, −6.45622455119055643020032056738, −4.95602613371290331412927020705, −4.52928013471042913641160867253, −2.44983989498102664965388830079, −1.85639763379586724715186269677, 2.02493035647338460071928648470, 2.65800957786767315766636269941, 4.13024855804980843881832128218, 5.49791380246749795125778916496, 6.57025184857760772981661569161, 7.36533485789274189151493668629, 8.593906660410840654170367728580, 9.277827842002824174897566236242, 10.10013671978383158058741984942, 10.88266940252495240352519544584

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