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Properties

Label 2-1428-119.81-c1-0-19
Degree $2$
Conductor $1428$
Sign $-0.236 + 0.971i$
Analytic cond. $11.4026$
Root an. cond. $3.37677$
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  + (−0.965 + 0.258i)3-s + (0.782 − 2.91i)5-s + (1.59 − 2.10i)7-s + (0.866 − 0.499i)9-s + (−0.0501 − 0.187i)11-s − 0.801·13-s + 3.02i·15-s + (2.64 − 3.16i)17-s + (2.42 − 1.40i)19-s + (−1 + 2.44i)21-s + (7.84 + 2.10i)23-s + (−3.58 − 2.06i)25-s + (−0.707 + 0.707i)27-s + (−1.08 − 1.08i)29-s + (−9.59 + 2.57i)31-s + ⋯
L(s)  = 1  + (−0.557 + 0.149i)3-s + (0.349 − 1.30i)5-s + (0.604 − 0.796i)7-s + (0.288 − 0.166i)9-s + (−0.0151 − 0.0564i)11-s − 0.222·13-s + 0.780i·15-s + (0.640 − 0.767i)17-s + (0.556 − 0.321i)19-s + (−0.218 + 0.534i)21-s + (1.63 + 0.438i)23-s + (−0.716 − 0.413i)25-s + (−0.136 + 0.136i)27-s + (−0.201 − 0.201i)29-s + (−1.72 + 0.461i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1428 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.236 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1428 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.236 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1428\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 17\)
Sign: $-0.236 + 0.971i$
Analytic conductor: \(11.4026\)
Root analytic conductor: \(3.37677\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1428} (1033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1428,\ (\ :1/2),\ -0.236 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.492667983\)
\(L(\frac12)\) \(\approx\) \(1.492667983\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (-1.59 + 2.10i)T \)
17 \( 1 + (-2.64 + 3.16i)T \)
good5 \( 1 + (-0.782 + 2.91i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.0501 + 0.187i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + 0.801T + 13T^{2} \)
19 \( 1 + (-2.42 + 1.40i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-7.84 - 2.10i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (1.08 + 1.08i)T + 29iT^{2} \)
31 \( 1 + (9.59 - 2.57i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-2.48 + 9.26i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (8.43 - 8.43i)T - 41iT^{2} \)
43 \( 1 + 1.24iT - 43T^{2} \)
47 \( 1 + (1.19 + 2.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.33 - 1.92i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-12.1 - 7.02i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (10.6 + 2.84i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.815 - 1.41i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.88 + 5.88i)T + 71iT^{2} \)
73 \( 1 + (3.23 - 0.867i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-11.2 - 3.01i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + 10.0iT - 83T^{2} \)
89 \( 1 + (3.68 + 6.38i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.0 + 13.0i)T + 97iT^{2} \)
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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

−9.289614562179699192510902608891, −8.696377724620677049369589601401, −7.52167241105543166921889603406, −7.08415961380257099345915975691, −5.63826783923848137527903640039, −5.13499765626992559424482601154, −4.50629916403552317868126485283, −3.32859996894947601572412770400, −1.60204008890475759685017853876, −0.69640284264849777162948278272, 1.56751239148562948722317290139, 2.64601719524199144288669241906, 3.60856845666960822117580005835, 5.05215392545639136436065685531, 5.62515919419158715746629171355, 6.54105983995215479368593619516, 7.19126037434959106149436364123, 8.048042478641098341852682719743, 9.025293601748444097839895393810, 9.929708874470406005505978390303

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