Properties

Label 2-456-152.101-c1-0-17
Degree $2$
Conductor $456$
Sign $0.858 + 0.513i$
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  + (−1.11 + 0.870i)2-s + (−0.342 + 0.939i)3-s + (0.483 − 1.94i)4-s + (−2.68 + 0.473i)5-s + (−0.437 − 1.34i)6-s + (0.416 − 0.720i)7-s + (1.15 + 2.58i)8-s + (−0.766 − 0.642i)9-s + (2.58 − 2.86i)10-s + (−1.90 + 1.09i)11-s + (1.65 + 1.11i)12-s + (−0.794 − 2.18i)13-s + (0.163 + 1.16i)14-s + (0.473 − 2.68i)15-s + (−3.53 − 1.87i)16-s + (1.80 − 1.51i)17-s + ⋯
L(s)  = 1  + (−0.787 + 0.615i)2-s + (−0.197 + 0.542i)3-s + (0.241 − 0.970i)4-s + (−1.20 + 0.211i)5-s + (−0.178 − 0.549i)6-s + (0.157 − 0.272i)7-s + (0.406 + 0.913i)8-s + (−0.255 − 0.214i)9-s + (0.816 − 0.906i)10-s + (−0.573 + 0.331i)11-s + (0.478 + 0.322i)12-s + (−0.220 − 0.605i)13-s + (0.0438 + 0.311i)14-s + (0.122 − 0.693i)15-s + (−0.883 − 0.469i)16-s + (0.438 − 0.368i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $0.858 + 0.513i$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ 0.858 + 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.481764 - 0.133180i\)
\(L(\frac12)\) \(\approx\) \(0.481764 - 0.133180i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.11 - 0.870i)T \)
3 \( 1 + (0.342 - 0.939i)T \)
19 \( 1 + (-3.91 + 1.90i)T \)
good5 \( 1 + (2.68 - 0.473i)T + (4.69 - 1.71i)T^{2} \)
7 \( 1 + (-0.416 + 0.720i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.90 - 1.09i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.794 + 2.18i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-1.80 + 1.51i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-1.13 + 6.45i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-1.10 + 1.31i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-2.45 + 4.25i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.34iT - 37T^{2} \)
41 \( 1 + (-7.26 - 2.64i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (1.29 - 0.229i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (4.64 + 3.89i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + (2.52 + 0.444i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (3.70 + 4.42i)T + (-10.2 + 58.1i)T^{2} \)
61 \( 1 + (1.61 + 0.284i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-10.2 + 12.2i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.67 + 9.50i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (7.83 + 2.85i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (-0.702 - 0.255i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (1.96 + 1.13i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.0971 + 0.0353i)T + (68.1 - 57.2i)T^{2} \)
97 \( 1 + (5.55 - 4.66i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.85228759495057255564613275947, −10.08006367066579511819707778827, −9.209553556868600476946520701586, −7.961473083183243104881271507047, −7.66877775253056030755827535128, −6.53457036036831082342109073806, −5.25717547994326927912685336141, −4.40266789523627428899201925574, −2.87324734718920327361915810756, −0.46293382504712002897280776671, 1.29261130543909106604695445975, 2.92773795040480658860326888472, 4.02391331907460446839630173101, 5.43542274388632341570555448521, 6.95033113510797336173259874903, 7.74403551499449520475785068447, 8.291699617255575855759885698416, 9.296731186527183140605458652604, 10.35754954157425098562087215091, 11.39649748614818750832895364874

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