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Properties

Label 2-456-456.11-c1-0-27
Degree $2$
Conductor $456$
Sign $0.876 - 0.481i$
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  + (0.389 − 1.35i)2-s + (0.845 + 1.51i)3-s + (−1.69 − 1.05i)4-s + (−0.277 + 0.479i)5-s + (2.38 − 0.561i)6-s + 2.13i·7-s + (−2.09 + 1.89i)8-s + (−1.57 + 2.55i)9-s + (0.544 + 0.563i)10-s + 2.58i·11-s + (0.165 − 3.46i)12-s + (3.72 − 2.15i)13-s + (2.90 + 0.832i)14-s + (−0.959 − 0.0131i)15-s + (1.75 + 3.59i)16-s + (4.54 + 2.62i)17-s + ⋯
L(s)  = 1  + (0.275 − 0.961i)2-s + (0.488 + 0.872i)3-s + (−0.848 − 0.529i)4-s + (−0.123 + 0.214i)5-s + (0.973 − 0.229i)6-s + 0.808i·7-s + (−0.742 + 0.670i)8-s + (−0.523 + 0.852i)9-s + (0.172 + 0.178i)10-s + 0.779i·11-s + (0.0476 − 0.998i)12-s + (1.03 − 0.596i)13-s + (0.777 + 0.222i)14-s + (−0.247 − 0.00339i)15-s + (0.439 + 0.898i)16-s + (1.10 + 0.635i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.876 - 0.481i$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ 0.876 - 0.481i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56152 + 0.400293i\)
\(L(\frac12)\) \(\approx\) \(1.56152 + 0.400293i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.389 + 1.35i)T \)
3 \( 1 + (-0.845 - 1.51i)T \)
19 \( 1 + (-0.400 - 4.34i)T \)
good5 \( 1 + (0.277 - 0.479i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 2.13iT - 7T^{2} \)
11 \( 1 - 2.58iT - 11T^{2} \)
13 \( 1 + (-3.72 + 2.15i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4.54 - 2.62i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (4.43 + 7.68i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.16 + 3.74i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.68iT - 31T^{2} \)
37 \( 1 + 3.91iT - 37T^{2} \)
41 \( 1 + (5.88 + 3.39i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.05 + 8.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.16 - 8.94i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.08 + 5.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.25 + 4.19i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.72 - 0.994i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.620 + 1.07i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.49 + 4.32i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.91 + 11.9i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.88 - 2.82i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.53iT - 83T^{2} \)
89 \( 1 + (-2.51 + 1.45i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.73 - 4.74i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.76184404036047220211741868731, −10.49749544078128086104428767679, −9.523401933412053885809716435756, −8.668508761589756058881782990251, −7.944720320513919840048293693228, −6.01642162373410884350558003134, −5.22427243184320705029668581061, −3.99572703195208749532664123849, −3.21777718453617354219928884616, −1.95695361767717694503539778605, 0.955651590888157800966695224247, 3.18437401262703931484322866412, 4.10802984138254936359334954379, 5.58502495156353956201027230790, 6.44700948055413718872177272139, 7.38755997037806809258000368468, 8.008928165152329834618981174601, 8.902122554891447518074770032472, 9.715015881095285076565875482085, 11.27752771456345310567288226502

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