Properties

Label 2-1014-13.9-c1-0-21
Degree $2$
Conductor $1014$
Sign $0.0128 + 0.999i$
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  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 5-s + (0.499 + 0.866i)6-s + (−1 − 1.73i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (1 − 1.73i)11-s − 0.999·12-s + 1.99·14-s + (0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−2.5 − 4.33i)17-s + 0.999·18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + 0.447·5-s + (0.204 + 0.353i)6-s + (−0.377 − 0.654i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (0.301 − 0.522i)11-s − 0.288·12-s + 0.534·14-s + (0.129 − 0.223i)15-s + (−0.125 + 0.216i)16-s + (−0.606 − 1.05i)17-s + 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.0128 + 0.999i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 0.0128 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.112626368\)
\(L(\frac12)\) \(\approx\) \(1.112626368\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−9.721598910888924831999700545300, −8.816132188809950993059637730209, −8.069065316441642037762337141329, −7.15518981365870002445049643724, −6.53720718255160533496112287902, −5.73313999863227770515650388184, −4.56111541945963310349183621560, −3.37441290502424608900821687074, −2.02225939261794840048945840586, −0.53502954448339628662698911697, 1.76070719377399213101038138835, 2.67801792064700889867737082458, 3.82911740763736640819164761892, 4.70398842287127940871627205330, 5.91120518933727436155557757813, 6.71422537800667986242028930894, 8.056804480285133042077098099725, 8.702572842985144090905953598584, 9.414001221970682274193897476387, 10.15083376395096151656682808912

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