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Properties

Label 2-456-456.11-c1-0-26
Degree $2$
Conductor $456$
Sign $0.237 - 0.971i$
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  + (1.36 + 0.372i)2-s + (−1.51 − 0.845i)3-s + (1.72 + 1.01i)4-s + (−0.450 + 0.779i)5-s + (−1.74 − 1.71i)6-s + 3.34i·7-s + (1.97 + 2.02i)8-s + (1.56 + 2.55i)9-s + (−0.905 + 0.896i)10-s − 2.25i·11-s + (−1.74 − 2.99i)12-s + (−4.95 + 2.86i)13-s + (−1.24 + 4.56i)14-s + (1.34 − 0.797i)15-s + (1.93 + 3.50i)16-s + (1.44 + 0.835i)17-s + ⋯
L(s)  = 1  + (0.964 + 0.263i)2-s + (−0.872 − 0.488i)3-s + (0.861 + 0.508i)4-s + (−0.201 + 0.348i)5-s + (−0.713 − 0.701i)6-s + 1.26i·7-s + (0.696 + 0.717i)8-s + (0.522 + 0.852i)9-s + (−0.286 + 0.283i)10-s − 0.679i·11-s + (−0.503 − 0.864i)12-s + (−1.37 + 0.793i)13-s + (−0.333 + 1.22i)14-s + (0.346 − 0.206i)15-s + (0.482 + 0.875i)16-s + (0.350 + 0.202i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.237 - 0.971i$
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: \((2,\ 456,\ (\ :1/2),\ 0.237 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39010 + 1.09072i\)
\(L(\frac12)\) \(\approx\) \(1.39010 + 1.09072i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.372i)T \)
3 \( 1 + (1.51 + 0.845i)T \)
19 \( 1 + (-4.11 + 1.42i)T \)
good5 \( 1 + (0.450 - 0.779i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.34iT - 7T^{2} \)
11 \( 1 + 2.25iT - 11T^{2} \)
13 \( 1 + (4.95 - 2.86i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.44 - 0.835i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.706 - 1.22i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.46 - 5.99i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.64iT - 31T^{2} \)
37 \( 1 - 1.11iT - 37T^{2} \)
41 \( 1 + (1.76 + 1.01i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.58 + 4.47i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.07 + 8.78i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.20 + 9.01i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.51 + 3.75i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-11.5 + 6.69i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.46 + 11.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.450 - 0.780i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.44 - 2.49i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.98 - 5.76i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 8.93iT - 83T^{2} \)
89 \( 1 + (-4.67 + 2.69i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.91 - 3.31i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.62839674820545370384358427423, −10.83369356354548246005130182469, −9.530772095381588843061858781766, −8.226457759245688407579257586006, −7.19858680764177146680707371977, −6.54250628555083649741813476588, −5.41507756069665000659441243965, −4.98307247516402124286557993141, −3.27880790965071667235659459512, −2.06962380063372273350724579335, 0.938478798821328192769251772687, 3.04621371970576189412904904265, 4.41588365883385366085932033929, 4.77021164616905756398076963397, 5.91772764428807587770487210517, 7.07184575277805554777038275577, 7.68631503402305505104660437311, 9.720881316427554263422620587302, 10.15061915991804084133161382881, 10.90231688318911923178653240278

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