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Properties

Label 2-780-12.11-c1-0-1
Degree $2$
Conductor $780$
Sign $-0.220 + 0.975i$
Analytic cond. $6.22833$
Root an. cond. $2.49566$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + (−1.68 − 0.382i)3-s − 2.00·4-s + i·5-s + (0.541 − 2.38i)6-s + 3.69i·7-s − 2.82i·8-s + (2.70 + 1.29i)9-s − 1.41·10-s − 4.01·11-s + (3.37 + 0.765i)12-s + 13-s − 5.22·14-s + (0.382 − 1.68i)15-s + 4.00·16-s + 4.82i·17-s + ⋯
L(s)  = 1  + 0.999i·2-s + (−0.975 − 0.220i)3-s − 1.00·4-s + 0.447i·5-s + (0.220 − 0.975i)6-s + 1.39i·7-s − 1.00i·8-s + (0.902 + 0.430i)9-s − 0.447·10-s − 1.20·11-s + (0.975 + 0.220i)12-s + 0.277·13-s − 1.39·14-s + (0.0988 − 0.436i)15-s + 1.00·16-s + 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(780\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 13\)
Sign: $-0.220 + 0.975i$
Analytic conductor: \(6.22833\)
Root analytic conductor: \(2.49566\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{780} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 780,\ (\ :1/2),\ -0.220 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.149069 - 0.186617i\)
\(L(\frac12)\) \(\approx\) \(0.149069 - 0.186617i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (1.68 + 0.382i)T \)
5 \( 1 - iT \)
13 \( 1 - T \)
good7 \( 1 - 3.69iT - 7T^{2} \)
11 \( 1 + 4.01T + 11T^{2} \)
17 \( 1 - 4.82iT - 17T^{2} \)
19 \( 1 - 0.317iT - 19T^{2} \)
23 \( 1 + 4.46T + 23T^{2} \)
29 \( 1 + 0.242iT - 29T^{2} \)
31 \( 1 + 5.99iT - 31T^{2} \)
37 \( 1 + 5.41T + 37T^{2} \)
41 \( 1 + 8.82iT - 41T^{2} \)
43 \( 1 + 3.37iT - 43T^{2} \)
47 \( 1 - 8.92T + 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 + 9.68T + 59T^{2} \)
61 \( 1 + 9.89T + 61T^{2} \)
67 \( 1 - 4.32iT - 67T^{2} \)
71 \( 1 + 8.79T + 71T^{2} \)
73 \( 1 - 9.41T + 73T^{2} \)
79 \( 1 + 0.634iT - 79T^{2} \)
83 \( 1 + 2.42T + 83T^{2} \)
89 \( 1 + 0.343iT - 89T^{2} \)
97 \( 1 + 5.65T + 97T^{2} \)
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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

−10.69942692647543300902049032064, −10.15953000187573019302254123750, −9.029685409781647522082124615640, −8.136663664092106595699760516426, −7.42663465935971349457310714333, −6.26668156079741661108269804706, −5.80277835578250252594093060803, −5.12147807566294332333818479782, −3.84940420722370650684332455313, −2.15120110335931636352653741089, 0.14605510976318197933286285618, 1.33596186247038664452295112721, 3.10784593284420447161654002342, 4.32447624553103532785563869782, 4.85578397481395936836071841391, 5.84452689277984456326157914123, 7.19550091209134941661773590152, 7.975773332920841653363428693065, 9.216761165989526966609542549803, 10.05370108593948913404932049314

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