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Properties

Label 2-1014-13.8-c2-0-30
Degree $2$
Conductor $1014$
Sign $0.697 + 0.716i$
Analytic cond. $27.6294$
Root an. cond. $5.25637$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s − 1.73·3-s + 2i·4-s + (4.73 + 4.73i)5-s + (1.73 + 1.73i)6-s + (−1.83 + 1.83i)7-s + (2 − 2i)8-s + 2.99·9-s − 9.46i·10-s + (10.7 − 10.7i)11-s − 3.46i·12-s + 3.66·14-s + (−8.19 − 8.19i)15-s − 4·16-s − 5.07i·17-s + (−2.99 − 2.99i)18-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s − 0.577·3-s + 0.5i·4-s + (0.946 + 0.946i)5-s + (0.288 + 0.288i)6-s + (−0.261 + 0.261i)7-s + (0.250 − 0.250i)8-s + 0.333·9-s − 0.946i·10-s + (0.975 − 0.975i)11-s − 0.288i·12-s + 0.261·14-s + (−0.546 − 0.546i)15-s − 0.250·16-s − 0.298i·17-s + (−0.166 − 0.166i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.697 + 0.716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.697 + 0.716i$
Analytic conductor: \(27.6294\)
Root analytic conductor: \(5.25637\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (775, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1),\ 0.697 + 0.716i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.400709380\)
\(L(\frac12)\) \(\approx\) \(1.400709380\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 + 1.73T \)
13 \( 1 \)
good5 \( 1 + (-4.73 - 4.73i)T + 25iT^{2} \)
7 \( 1 + (1.83 - 1.83i)T - 49iT^{2} \)
11 \( 1 + (-10.7 + 10.7i)T - 121iT^{2} \)
17 \( 1 + 5.07iT - 289T^{2} \)
19 \( 1 + (-2.60 - 2.60i)T + 361iT^{2} \)
23 \( 1 + 31.8iT - 529T^{2} \)
29 \( 1 - 17.3T + 841T^{2} \)
31 \( 1 + (30.8 + 30.8i)T + 961iT^{2} \)
37 \( 1 + (-36.1 + 36.1i)T - 1.36e3iT^{2} \)
41 \( 1 + (30.2 + 30.2i)T + 1.68e3iT^{2} \)
43 \( 1 - 32.6iT - 1.84e3T^{2} \)
47 \( 1 + (58.9 - 58.9i)T - 2.20e3iT^{2} \)
53 \( 1 - 97.6T + 2.80e3T^{2} \)
59 \( 1 + (-43.2 + 43.2i)T - 3.48e3iT^{2} \)
61 \( 1 - 79.3T + 3.72e3T^{2} \)
67 \( 1 + (46.7 + 46.7i)T + 4.48e3iT^{2} \)
71 \( 1 + (-37.9 - 37.9i)T + 5.04e3iT^{2} \)
73 \( 1 + (-27.5 + 27.5i)T - 5.32e3iT^{2} \)
79 \( 1 - 36.1T + 6.24e3T^{2} \)
83 \( 1 + (20.9 + 20.9i)T + 6.88e3iT^{2} \)
89 \( 1 + (-49.3 + 49.3i)T - 7.92e3iT^{2} \)
97 \( 1 + (12.9 + 12.9i)T + 9.40e3iT^{2} \)
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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

−9.700714527329174156050477700464, −9.142955464155694314793790552600, −8.119927993318255789711962369348, −6.90497570561933363498398952433, −6.32789033615146278952566728580, −5.61047054550105183373096548613, −4.16904085749451853741110990517, −3.05004990556356021220342301709, −2.10423878493959468922047560472, −0.68631695313620981812440112279, 1.05478959771780063357192154388, 1.86045446992436044702060833195, 3.83964505600091936448453797234, 4.99960001746685316627693558055, 5.53246281062637806100895719660, 6.61373998841638400408864240095, 7.10127268737276861693950359793, 8.352099592277214150311032793331, 9.145359601307399624580882445007, 9.826649354567412725407250437412

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