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Properties

Label 2-1568-4.3-c2-0-7
Degree $2$
Conductor $1568$
Sign $-0.707 + 0.707i$
Analytic cond. $42.7249$
Root an. cond. $6.53642$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.55i·3-s − 9.86·5-s + 2.47·9-s + 13.1i·11-s + 5.86·13-s − 25.2i·15-s + 0.570·17-s + 15.6i·19-s + 16.4i·23-s + 72.3·25-s + 29.3i·27-s − 29.7·29-s + 54.8i·31-s − 33.6·33-s − 42.0·37-s + ⋯
L(s)  = 1  + 0.851i·3-s − 1.97·5-s + 0.274·9-s + 1.19i·11-s + 0.451·13-s − 1.68i·15-s + 0.0335·17-s + 0.824i·19-s + 0.716i·23-s + 2.89·25-s + 1.08i·27-s − 1.02·29-s + 1.76i·31-s − 1.01·33-s − 1.13·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(42.7249\)
Root analytic conductor: \(6.53642\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (1471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5301644047\)
\(L(\frac12)\) \(\approx\) \(0.5301644047\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 - 2.55iT - 9T^{2} \)
5 \( 1 + 9.86T + 25T^{2} \)
11 \( 1 - 13.1iT - 121T^{2} \)
13 \( 1 - 5.86T + 169T^{2} \)
17 \( 1 - 0.570T + 289T^{2} \)
19 \( 1 - 15.6iT - 361T^{2} \)
23 \( 1 - 16.4iT - 529T^{2} \)
29 \( 1 + 29.7T + 841T^{2} \)
31 \( 1 - 54.8iT - 961T^{2} \)
37 \( 1 + 42.0T + 1.36e3T^{2} \)
41 \( 1 + 0.773T + 1.68e3T^{2} \)
43 \( 1 - 41.7iT - 1.84e3T^{2} \)
47 \( 1 + 58.4iT - 2.20e3T^{2} \)
53 \( 1 - 5.65T + 2.80e3T^{2} \)
59 \( 1 + 42.6iT - 3.48e3T^{2} \)
61 \( 1 - 95.9T + 3.72e3T^{2} \)
67 \( 1 - 69.8iT - 4.48e3T^{2} \)
71 \( 1 + 92.0iT - 5.04e3T^{2} \)
73 \( 1 + 9.97T + 5.32e3T^{2} \)
79 \( 1 + 20.1iT - 6.24e3T^{2} \)
83 \( 1 + 151. iT - 6.88e3T^{2} \)
89 \( 1 + 5.79T + 7.92e3T^{2} \)
97 \( 1 + 103.T + 9.40e3T^{2} \)
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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

−9.819529743696355172624010719819, −8.882224993236417880428720604846, −8.170193576997574387132409330131, −7.31558081754042891954857480924, −6.88983602307710405973188091172, −5.28437428298534302312786529418, −4.58608647007620106916905150527, −3.79934272751278575812573805990, −3.36642102694472761126565300528, −1.52218230523687537302806510728, 0.18864055044335757825032908649, 0.970464510385337210719389483255, 2.60577854955976573794306809398, 3.71306800653885188733933195123, 4.24722138482820842545023995457, 5.49694601342697044152885543286, 6.63955072751853372783933806299, 7.18032125674073773625856703941, 8.011475631503717762023643486754, 8.377054225807315467997627136792

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