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Properties

Label 2-1014-13.10-c1-0-0
Degree $2$
Conductor $1014$
Sign $-0.982 - 0.188i$
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 − 3.15i·5-s + (−0.866 − 0.499i)6-s + (−4.06 − 2.34i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (1.57 + 2.73i)10-s + (−0.118 + 0.0685i)11-s + 0.999·12-s + 4.69·14-s + (2.73 − 1.57i)15-s + (−0.5 − 0.866i)16-s + (−2.80 + 4.85i)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 − 1.41i·5-s + (−0.353 − 0.204i)6-s + (−1.53 − 0.886i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.499 + 0.865i)10-s + (−0.0357 + 0.0206i)11-s + 0.288·12-s + 1.25·14-s + (0.706 − 0.407i)15-s + (−0.125 − 0.216i)16-s + (−0.679 + 1.17i)17-s − 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.982 - 0.188i$
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: \((2,\ 1014,\ (\ :1/2),\ -0.982 - 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.02681688127\)
\(L(\frac12)\) \(\approx\) \(0.02681688127\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.15iT - 5T^{2} \)
7 \( 1 + (4.06 + 2.34i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.118 - 0.0685i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.80 - 4.85i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.31 - 2.49i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.04 + 5.28i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.425 - 0.736i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.23iT - 31T^{2} \)
37 \( 1 + (10.1 - 5.85i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.70 - 2.13i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.04 - 1.81i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.98iT - 47T^{2} \)
53 \( 1 + 1.82T + 53T^{2} \)
59 \( 1 + (5.10 + 2.94i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.19 - 3.80i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.08 - 2.35i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.0847 - 0.0489i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.32iT - 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 + 9.85iT - 83T^{2} \)
89 \( 1 + (14.7 - 8.54i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.84 + 1.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.15508484921238960184398481985, −9.489544874279670545134569839468, −8.681098319859999542661405150491, −8.187260712970402797852527209829, −6.98792652755107831712697952446, −6.23213295246660541147270407517, −5.15279748540686141592551919245, −4.18995027968689802911378558569, −3.24930830927308384512608542227, −1.45796610040519098674853237700, 0.01406836951807021542236876520, 2.21895523008014369296045943954, 2.93777996934053882708910019967, 3.56951160683934520963424759482, 5.55093904431307047003920309786, 6.50936371613538573858215814601, 7.04354058512577006466984726635, 7.75649132633690695038298867217, 9.118824227465446970400149239595, 9.426286956143857188307460006482

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