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Properties

Label 2-1428-17.16-c1-0-6
Degree $2$
Conductor $1428$
Sign $0.668 - 0.743i$
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  − i·3-s + 0.981i·5-s + i·7-s − 9-s + 0.478i·11-s − 6.19·13-s + 0.981·15-s + (2.75 − 3.06i)17-s + 6.75·19-s + 21-s + 6.21i·23-s + 4.03·25-s + i·27-s + 1.43i·29-s + 2.44i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.439i·5-s + 0.377i·7-s − 0.333·9-s + 0.144i·11-s − 1.71·13-s + 0.253·15-s + (0.668 − 0.743i)17-s + 1.55·19-s + 0.218·21-s + 1.29i·23-s + 0.807·25-s + 0.192i·27-s + 0.266i·29-s + 0.438i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1428 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1428 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1428\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 17\)
Sign: $0.668 - 0.743i$
Analytic conductor: \(11.4026\)
Root analytic conductor: \(3.37677\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1428} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1428,\ (\ :1/2),\ 0.668 - 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.408452648\)
\(L(\frac12)\) \(\approx\) \(1.408452648\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
7 \( 1 - iT \)
17 \( 1 + (-2.75 + 3.06i)T \)
good5 \( 1 - 0.981iT - 5T^{2} \)
11 \( 1 - 0.478iT - 11T^{2} \)
13 \( 1 + 6.19T + 13T^{2} \)
19 \( 1 - 6.75T + 19T^{2} \)
23 \( 1 - 6.21iT - 23T^{2} \)
29 \( 1 - 1.43iT - 29T^{2} \)
31 \( 1 - 2.44iT - 31T^{2} \)
37 \( 1 - 10.1iT - 37T^{2} \)
41 \( 1 - 7.68iT - 41T^{2} \)
43 \( 1 + 0.741T + 43T^{2} \)
47 \( 1 - 2.44T + 47T^{2} \)
53 \( 1 - 13.2T + 53T^{2} \)
59 \( 1 + 7.45T + 59T^{2} \)
61 \( 1 - 0.680iT - 61T^{2} \)
67 \( 1 + 15.0T + 67T^{2} \)
71 \( 1 + 3.73iT - 71T^{2} \)
73 \( 1 - 7.17iT - 73T^{2} \)
79 \( 1 - 13.2iT - 79T^{2} \)
83 \( 1 - 9.34T + 83T^{2} \)
89 \( 1 - 17.7T + 89T^{2} \)
97 \( 1 + 3.24iT - 97T^{2} \)
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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

−9.699804673982980821201983037475, −8.922336947677892192764178295541, −7.64185957945382176234880377637, −7.42719474395609634264350132147, −6.54212861094989108793229651700, −5.38319760371634906041066070973, −4.89930533407482713777943337503, −3.22838677773018617045740736568, −2.65834254654897008937867332819, −1.25287159382474223558253231264, 0.62566977786002727702878360175, 2.31306355514772558350478797950, 3.41964586237747854296106407498, 4.43710212019365410141176705198, 5.14197578596333877793192101248, 5.94742424004033096518048355276, 7.23155058963404800242703332682, 7.71387174140338587333798640381, 8.837973994572954620236603101805, 9.400697648245310765957153299183

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