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Properties

Label 2-1568-7.3-c0-0-2
Degree $2$
Conductor $1568$
Sign $0.769 + 0.638i$
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  + (1.60 − 0.923i)5-s + (−0.5 − 0.866i)9-s + 0.765i·13-s + (0.662 + 0.382i)17-s + (1.20 − 2.09i)25-s − 1.41·29-s + (−0.707 − 1.22i)37-s + 1.84i·41-s + (−1.60 − 0.923i)45-s + (−0.662 + 0.382i)61-s + (0.707 + 1.22i)65-s + (1.60 + 0.923i)73-s + (−0.499 + 0.866i)81-s + 1.41·85-s + (0.662 − 0.382i)89-s + ⋯
L(s)  = 1  + (1.60 − 0.923i)5-s + (−0.5 − 0.866i)9-s + 0.765i·13-s + (0.662 + 0.382i)17-s + (1.20 − 2.09i)25-s − 1.41·29-s + (−0.707 − 1.22i)37-s + 1.84i·41-s + (−1.60 − 0.923i)45-s + (−0.662 + 0.382i)61-s + (0.707 + 1.22i)65-s + (1.60 + 0.923i)73-s + (−0.499 + 0.866i)81-s + 1.41·85-s + (0.662 − 0.382i)89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.769 + 0.638i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :0),\ 0.769 + 0.638i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.406729794\)
\(L(\frac12)\) \(\approx\) \(1.406729794\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (-1.60 + 0.923i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - 0.765iT - T^{2} \)
17 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - 1.84iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.60 - 0.923i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.662 + 0.382i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + 1.84iT - T^{2} \)
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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

−9.415980778577028903618206006042, −9.013466129936809320052827080885, −8.166993682311061136395112279939, −6.95006393427127460323440548619, −6.05052101872747675007691464804, −5.63270862132446842326439392614, −4.67504163049824262980286220374, −3.54721708422631498170683643710, −2.24730351447737267421377159830, −1.28551354866079674503287280046, 1.76926603684692060658408155464, 2.62100399941984294837764905063, 3.46860577906764404918639055609, 5.19453018661799532519517491914, 5.51455860669581696237758368941, 6.39766376471209992981287604288, 7.26428464222766024350464097142, 8.050042202237634706648817584869, 9.123747295716281988305147941117, 9.754525047155074749008036635382

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