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Properties

Label 2-1014-13.10-c1-0-6
Degree $2$
Conductor $1014$
Sign $0.0502 - 0.998i$
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 + 2.13i·5-s + (0.866 + 0.499i)6-s + (−0.0423 − 0.0244i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (1.06 + 1.85i)10-s + (−5.45 + 3.14i)11-s + 0.999·12-s − 0.0489·14-s + (−1.85 + 1.06i)15-s + (−0.5 − 0.866i)16-s + (−1.44 + 2.50i)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 + 0.955i·5-s + (0.353 + 0.204i)6-s + (−0.0160 − 0.00924i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.337 + 0.585i)10-s + (−1.64 + 0.949i)11-s + 0.288·12-s − 0.0130·14-s + (−0.477 + 0.275i)15-s + (−0.125 − 0.216i)16-s + (−0.350 + 0.607i)17-s + 0.235i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0502 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0502 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.0502 - 0.998i$
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: \((2,\ 1014,\ (\ :1/2),\ 0.0502 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.021626303\)
\(L(\frac12)\) \(\approx\) \(2.021626303\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.13iT - 5T^{2} \)
7 \( 1 + (0.0423 + 0.0244i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.45 - 3.14i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.44 - 2.50i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-6.24 - 3.60i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.35 - 2.35i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.45 + 4.25i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 9.00iT - 31T^{2} \)
37 \( 1 + (-0.152 + 0.0881i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (7.44 - 4.29i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.35 + 5.81i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.20iT - 47T^{2} \)
53 \( 1 - 9.34T + 53T^{2} \)
59 \( 1 + (3.69 + 2.13i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.55 - 6.15i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.66 + 2.69i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.54 - 4.35i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 14.9iT - 73T^{2} \)
79 \( 1 - 13.8T + 79T^{2} \)
83 \( 1 + 11.1iT - 83T^{2} \)
89 \( 1 + (3.39 - 1.96i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.14 - 1.23i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.21633337448693720381985931662, −9.787595274144263525107480157916, −8.469555465346722596934125379527, −7.51185109846097455307815103061, −6.85035315025494328962840795771, −5.54682469365731836410010028937, −4.98267527000656188126593208264, −3.72368186976624234665034579935, −2.97833382222493824503443059479, −1.99713997687850451049693022598, 0.70315360204804764395148771796, 2.47726315999522166539472169093, 3.31749665213744930156539310969, 4.80522232789633237452392497890, 5.27746281057782607017755606141, 6.23349789830970398555570777530, 7.40912916814144369591992324301, 7.904354521172609905907965565202, 8.799753584150705298311746019055, 9.471662938381647429178242995658

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