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Properties

Label 2-1568-56.51-c0-0-1
Degree $2$
Conductor $1568$
Sign $-0.749 + 0.661i$
Analytic cond. $0.782533$
Root an. cond. $0.884609$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 1.22i)3-s + (−0.499 + 0.866i)9-s + (−0.707 − 1.22i)17-s + (0.707 − 1.22i)19-s + (−0.5 − 0.866i)25-s − 1.41·41-s + (−0.999 + 1.73i)51-s − 2·57-s + (0.707 + 1.22i)59-s + (−1 − 1.73i)67-s + (0.707 + 1.22i)73-s + (−0.707 + 1.22i)75-s + (0.499 + 0.866i)81-s − 1.41·83-s + (0.707 − 1.22i)89-s + ⋯
L(s)  = 1  + (−0.707 − 1.22i)3-s + (−0.499 + 0.866i)9-s + (−0.707 − 1.22i)17-s + (0.707 − 1.22i)19-s + (−0.5 − 0.866i)25-s − 1.41·41-s + (−0.999 + 1.73i)51-s − 2·57-s + (0.707 + 1.22i)59-s + (−1 − 1.73i)67-s + (0.707 + 1.22i)73-s + (−0.707 + 1.22i)75-s + (0.499 + 0.866i)81-s − 1.41·83-s + (0.707 − 1.22i)89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $-0.749 + 0.661i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :0),\ -0.749 + 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7048894388\)
\(L(\frac12)\) \(\approx\) \(0.7048894388\)
\(L(1)\) 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 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−9.261653931770494779331088174037, −8.424061565339336070774882424436, −7.40023798192802944056116222381, −6.98444390120170911501123707020, −6.22395263508943469269845695649, −5.32202424268474522195061305823, −4.49732426103536543827852661755, −3.00960965875314114017386653947, −1.97634445909428940020529178537, −0.60785572555413721874552331542, 1.77387166347442356248440710328, 3.44183551224433103052156773312, 4.03528631296261821245245372936, 5.01653302245789992433759287189, 5.69808705547660557947827031274, 6.45926908840564998333299969446, 7.59666000456584252244920764689, 8.477129063328749820104495902792, 9.329462768526066084049652089046, 10.10370823123184828174894224212

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