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Properties

Label 2-776-776.3-c0-0-0
Degree $2$
Conductor $776$
Sign $0.687 + 0.726i$
Analytic cond. $0.387274$
Root an. cond. $0.622313$
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  + (0.793 − 0.608i)2-s + (1.20 + 0.158i)3-s + (0.258 − 0.965i)4-s + (1.05 − 0.608i)6-s + (−0.382 − 0.923i)8-s + (0.465 + 0.124i)9-s + (−1.17 + 1.53i)11-s + (0.465 − 1.12i)12-s + (−0.866 − 0.499i)16-s + (−1.57 + 0.534i)17-s + (0.445 − 0.184i)18-s + (0.382 + 0.0761i)19-s + 1.93i·22-s + (−0.315 − 1.17i)24-s + (−0.130 − 0.991i)25-s + ⋯
L(s)  = 1  + (0.793 − 0.608i)2-s + (1.20 + 0.158i)3-s + (0.258 − 0.965i)4-s + (1.05 − 0.608i)6-s + (−0.382 − 0.923i)8-s + (0.465 + 0.124i)9-s + (−1.17 + 1.53i)11-s + (0.465 − 1.12i)12-s + (−0.866 − 0.499i)16-s + (−1.57 + 0.534i)17-s + (0.445 − 0.184i)18-s + (0.382 + 0.0761i)19-s + 1.93i·22-s + (−0.315 − 1.17i)24-s + (−0.130 − 0.991i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(776\)    =    \(2^{3} \cdot 97\)
Sign: $0.687 + 0.726i$
Analytic conductor: \(0.387274\)
Root analytic conductor: \(0.622313\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{776} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 776,\ (\ :0),\ 0.687 + 0.726i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.885726359\)
\(L(\frac12)\) \(\approx\) \(1.885726359\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.793 + 0.608i)T \)
97 \( 1 + (0.991 + 0.130i)T \)
good3 \( 1 + (-1.20 - 0.158i)T + (0.965 + 0.258i)T^{2} \)
5 \( 1 + (0.130 + 0.991i)T^{2} \)
7 \( 1 + (0.793 + 0.608i)T^{2} \)
11 \( 1 + (1.17 - 1.53i)T + (-0.258 - 0.965i)T^{2} \)
13 \( 1 + (-0.130 - 0.991i)T^{2} \)
17 \( 1 + (1.57 - 0.534i)T + (0.793 - 0.608i)T^{2} \)
19 \( 1 + (-0.382 - 0.0761i)T + (0.923 + 0.382i)T^{2} \)
23 \( 1 + (-0.608 - 0.793i)T^{2} \)
29 \( 1 + (0.991 - 0.130i)T^{2} \)
31 \( 1 + (0.965 + 0.258i)T^{2} \)
37 \( 1 + (0.608 - 0.793i)T^{2} \)
41 \( 1 + (-1.95 + 0.128i)T + (0.991 - 0.130i)T^{2} \)
43 \( 1 + (-1.91 + 0.513i)T + (0.866 - 0.5i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (0.258 - 0.965i)T^{2} \)
59 \( 1 + (-0.996 + 0.491i)T + (0.608 - 0.793i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.389 - 1.95i)T + (-0.923 - 0.382i)T^{2} \)
71 \( 1 + (-0.991 - 0.130i)T^{2} \)
73 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (-0.707 + 0.707i)T^{2} \)
83 \( 1 + (-0.284 - 0.837i)T + (-0.793 + 0.608i)T^{2} \)
89 \( 1 + (0.0999 + 0.241i)T + (-0.707 + 0.707i)T^{2} \)
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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

−10.37506340195342271088017780469, −9.661611573653055113796866933518, −8.898295517187167857588365182095, −7.84739604707887270184900883817, −6.98383479876779012654954140603, −5.79342723047607501595368617593, −4.60777478978195990022107760381, −3.99134507712998496982844107864, −2.59016389294132225993947804094, −2.17731098216373787259311447202, 2.49248839389802738513101195040, 3.05198291454220799478992853258, 4.15967702557413684417487663688, 5.33837409424338983171424892472, 6.16511399860825639572109228871, 7.39081029398081604737716565546, 7.910698791724956021441988253258, 8.764985194178553633026444151879, 9.303616162386302643764588628430, 11.00323880985023812405810529617

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