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Properties

Label 2-1014-13.10-c1-0-21
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 − 3i·5-s + (−0.866 − 0.499i)6-s + (−1.73 − i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−1.5 − 2.59i)10-s + (5.19 − 3i)11-s − 0.999·12-s − 1.99·14-s + (−2.59 + 1.5i)15-s + (−0.5 − 0.866i)16-s + (−1.5 + 2.59i)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 − 1.34i·5-s + (−0.353 − 0.204i)6-s + (−0.654 − 0.377i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.474 − 0.821i)10-s + (1.56 − 0.904i)11-s − 0.288·12-s − 0.534·14-s + (−0.670 + 0.387i)15-s + (−0.125 − 0.216i)16-s + (−0.363 + 0.630i)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

\(L(1)\) \(\approx\) \(1.807236470\)
\(L(\frac12)\) \(\approx\) \(1.807236470\)
\(L(\frac{3}{2})\) 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 \( 1 + 3iT - 5T^{2} \)
7 \( 1 + (1.73 + i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.19 + 3i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
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 \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + (6.06 - 3.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.59 + 1.5i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (5 - 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6iT - 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.66 + 5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.19 - 3i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 13iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (-15.5 + 9i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-12.1 - 7i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.541225755237084765139298462973, −8.794745303995626453140458688325, −8.030411549403024073379767282660, −6.58481587249752963009041858118, −6.26497731389346805362335782377, −5.16001363357892419998338734187, −4.22181612132743795954209643813, −3.39465505802359703649597725929, −1.70551341016241489158102838074, −0.71413539558051888271713312617, 2.19862481020671240957697544092, 3.40825892294501027524953994898, 3.97924355170981940171333997553, 5.24054533123160526874042537937, 6.15819641573394723614536024220, 6.90786262156017887196013737128, 7.30904608319009807908097424965, 8.867597722086784572459645357819, 9.604261024892860827481512682516, 10.26486032877083964542074925373

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