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Properties

Label 2-776-776.163-c0-0-0
Degree $2$
Conductor $776$
Sign $-0.996 - 0.0809i$
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  + (0.608 + 0.793i)2-s + (−0.207 + 1.57i)3-s + (−0.258 + 0.965i)4-s + (−1.37 + 0.793i)6-s + (−0.923 + 0.382i)8-s + (−1.46 − 0.392i)9-s + (−1.53 − 1.17i)11-s + (−1.46 − 0.607i)12-s + (−0.866 − 0.499i)16-s + (1.75 + 0.867i)17-s + (−0.580 − 1.40i)18-s + (0.923 + 1.38i)19-s − 1.93i·22-s + (−0.410 − 1.53i)24-s + (0.991 − 0.130i)25-s + ⋯
L(s)  = 1  + (0.608 + 0.793i)2-s + (−0.207 + 1.57i)3-s + (−0.258 + 0.965i)4-s + (−1.37 + 0.793i)6-s + (−0.923 + 0.382i)8-s + (−1.46 − 0.392i)9-s + (−1.53 − 1.17i)11-s + (−1.46 − 0.607i)12-s + (−0.866 − 0.499i)16-s + (1.75 + 0.867i)17-s + (−0.580 − 1.40i)18-s + (0.923 + 1.38i)19-s − 1.93i·22-s + (−0.410 − 1.53i)24-s + (0.991 − 0.130i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(776\)    =    \(2^{3} \cdot 97\)
Sign: $-0.996 - 0.0809i$
Analytic conductor: \(0.387274\)
Root analytic conductor: \(0.622313\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{776} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 776,\ (\ :0),\ -0.996 - 0.0809i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.080612978\)
\(L(\frac12)\) \(\approx\) \(1.080612978\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.608 - 0.793i)T \)
97 \( 1 + (0.130 - 0.991i)T \)
good3 \( 1 + (0.207 - 1.57i)T + (-0.965 - 0.258i)T^{2} \)
5 \( 1 + (-0.991 + 0.130i)T^{2} \)
7 \( 1 + (0.608 - 0.793i)T^{2} \)
11 \( 1 + (1.53 + 1.17i)T + (0.258 + 0.965i)T^{2} \)
13 \( 1 + (0.991 - 0.130i)T^{2} \)
17 \( 1 + (-1.75 - 0.867i)T + (0.608 + 0.793i)T^{2} \)
19 \( 1 + (-0.923 - 1.38i)T + (-0.382 + 0.923i)T^{2} \)
23 \( 1 + (0.793 - 0.608i)T^{2} \)
29 \( 1 + (0.130 + 0.991i)T^{2} \)
31 \( 1 + (-0.965 - 0.258i)T^{2} \)
37 \( 1 + (-0.793 - 0.608i)T^{2} \)
41 \( 1 + (0.835 + 0.732i)T + (0.130 + 0.991i)T^{2} \)
43 \( 1 + (0.252 - 0.0675i)T + (0.866 - 0.5i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-0.258 + 0.965i)T^{2} \)
59 \( 1 + (0.125 - 0.369i)T + (-0.793 - 0.608i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1.25 + 0.835i)T + (0.382 - 0.923i)T^{2} \)
71 \( 1 + (-0.130 + 0.991i)T^{2} \)
73 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
79 \( 1 + (0.707 - 0.707i)T^{2} \)
83 \( 1 + (-0.837 + 1.69i)T + (-0.608 - 0.793i)T^{2} \)
89 \( 1 + (-1.83 + 0.758i)T + (0.707 - 0.707i)T^{2} \)
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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

−10.63043274004472612949669617561, −10.25168816817921955250840268314, −9.202608171093822100922624478681, −8.208533309874924860943390082035, −7.72369537035723680812925576289, −6.07115703306754285941865361036, −5.48324041142169485806790316803, −4.89312272863898396945822905607, −3.53828698242151537928028170795, −3.21226345030986740787971304083, 1.02491095737652947525179142036, 2.34982065203221303377489236118, 3.09518300489719053938996253025, 5.02755959479523925376599292194, 5.33251611452621329307860215884, 6.72607309250042065918367872427, 7.32132635275482893534813953049, 8.169625496266847879897372315729, 9.529641102328282497998129073012, 10.20601502282304639177832780589

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