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Properties

Label 2-1014-13.10-c1-0-8
Degree $2$
Conductor $1014$
Sign $0.964 - 0.265i$
Analytic cond. $8.09683$
Root an. cond. $2.84549$
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  + (−0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s − i·5-s + (−0.866 − 0.499i)6-s + (−1.73 − i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)10-s + (−1.73 + i)11-s + 0.999·12-s + 1.99·14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (2.5 − 4.33i)17-s − 0.999i·18-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s − 0.447i·5-s + (−0.353 − 0.204i)6-s + (−0.654 − 0.377i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.158 + 0.273i)10-s + (−0.522 + 0.301i)11-s + 0.288·12-s + 0.534·14-s + (0.223 − 0.129i)15-s + (−0.125 − 0.216i)16-s + (0.606 − 1.05i)17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.964 - 0.265i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 0.964 - 0.265i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.187054906\)
\(L(\frac12)\) \(\approx\) \(1.187054906\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 \)
good5 \( 1 + iT - 5T^{2} \)
7 \( 1 + (1.73 + i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.73 - i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.73 - i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 + (-9.52 + 5.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.33 + 2.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 2iT - 47T^{2} \)
53 \( 1 + T + 53T^{2} \)
59 \( 1 + (6.92 + 4i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.5 + 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.73 + i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-12.1 - 7i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 13iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (1.73 - i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.73 - i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.613324088267439173904891936223, −9.430381559247759881811649050040, −8.401826004571441718630804443673, −7.50798041115332066387866312838, −6.89814275167028823054322365165, −5.57628344967917593115606222469, −4.95282952159382361517300713027, −3.66492497596857318336334144592, −2.60120632224401171121048076864, −0.861688325319878772331787383242, 1.00442739907000154704838934747, 2.64853706010199820781830749965, 3.04908599220719174876132770834, 4.50207464573377221875339068431, 6.04628845460694285616243849823, 6.51915324578637967625520898980, 7.65886547512134689192113437221, 8.231198761403432099835121411187, 9.109380144882419430297784277559, 9.908819878671511376085753358904

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