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Properties

Label 2-456-456.11-c1-0-57
Degree $2$
Conductor $456$
Sign $0.927 + 0.372i$
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  + (1.39 + 0.212i)2-s + (−1.43 + 0.968i)3-s + (1.91 + 0.592i)4-s + (1.07 − 1.86i)5-s + (−2.21 + 1.05i)6-s − 3.66i·7-s + (2.54 + 1.23i)8-s + (1.12 − 2.78i)9-s + (1.90 − 2.38i)10-s − 3.82i·11-s + (−3.31 + 0.999i)12-s + (−1.79 + 1.03i)13-s + (0.777 − 5.12i)14-s + (0.261 + 3.72i)15-s + (3.29 + 2.26i)16-s + (3.42 + 1.98i)17-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)2-s + (−0.828 + 0.559i)3-s + (0.955 + 0.296i)4-s + (0.482 − 0.835i)5-s + (−0.903 + 0.428i)6-s − 1.38i·7-s + (0.899 + 0.436i)8-s + (0.374 − 0.927i)9-s + (0.602 − 0.753i)10-s − 1.15i·11-s + (−0.957 + 0.288i)12-s + (−0.497 + 0.287i)13-s + (0.207 − 1.37i)14-s + (0.0675 + 0.962i)15-s + (0.824 + 0.566i)16-s + (0.831 + 0.480i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 + 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.927 + 0.372i$
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: \((2,\ 456,\ (\ :1/2),\ 0.927 + 0.372i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11178 - 0.408420i\)
\(L(\frac12)\) \(\approx\) \(2.11178 - 0.408420i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 - 0.212i)T \)
3 \( 1 + (1.43 - 0.968i)T \)
19 \( 1 + (4.15 - 1.33i)T \)
good5 \( 1 + (-1.07 + 1.86i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.66iT - 7T^{2} \)
11 \( 1 + 3.82iT - 11T^{2} \)
13 \( 1 + (1.79 - 1.03i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.42 - 1.98i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.581 - 1.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.45 - 5.98i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.31iT - 31T^{2} \)
37 \( 1 + 8.74iT - 37T^{2} \)
41 \( 1 + (-1.80 - 1.04i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.31 - 9.21i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.09 + 5.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.84 - 6.65i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.250 + 0.144i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (9.24 - 5.33i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.31 - 4.01i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.99 + 8.64i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.73 + 9.93i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.33 - 1.34i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.7iT - 83T^{2} \)
89 \( 1 + (-2.48 + 1.43i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.13 - 14.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.87473042810858846495867465138, −10.57588128306455827205324120098, −9.417535209538919268248850056282, −8.138392265106449270545198839549, −6.97545160024476153370030227904, −6.05990343006833623024669835080, −5.22362307540285281602345582694, −4.33527070151716402728800966701, −3.46000578318751223699761861049, −1.21548362222489507992548410260, 2.04130041781298558147396904478, 2.75370438389199016944552114009, 4.67703216689790343074876974044, 5.44428879129748076569662192000, 6.37625739294343820800500994508, 6.94192020681939808007916579596, 8.088263129874878081473017418767, 9.807573959356134287870700886614, 10.37767453249563710603335733164, 11.44046553613264965255875955570

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