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Properties

Label 2-1014-13.5-c2-0-25
Degree $2$
Conductor $1014$
Sign $0.208 - 0.977i$
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 + (1.26 − 1.26i)5-s + (−1.73 + 1.73i)6-s + (6.83 + 6.83i)7-s + (2 + 2i)8-s + 2.99·9-s + 2.53i·10-s + (7.26 + 7.26i)11-s − 3.46i·12-s − 13.6·14-s + (2.19 − 2.19i)15-s − 4·16-s + 18.9i·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 + (0.253 − 0.253i)5-s + (−0.288 + 0.288i)6-s + (0.975 + 0.975i)7-s + (0.250 + 0.250i)8-s + 0.333·9-s + 0.253i·10-s + (0.660 + 0.660i)11-s − 0.288i·12-s − 0.975·14-s + (0.146 − 0.146i)15-s − 0.250·16-s + 1.11i·17-s + (−0.166 + 0.166i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 - 0.977i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.208 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.208 - 0.977i$
Analytic conductor: \(27.6294\)
Root analytic conductor: \(5.25637\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1),\ 0.208 - 0.977i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.262762162\)
\(L(\frac12)\) \(\approx\) \(2.262762162\)
\(L(2)\) 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 \( 1 + (-1.26 + 1.26i)T - 25iT^{2} \)
7 \( 1 + (-6.83 - 6.83i)T + 49iT^{2} \)
11 \( 1 + (-7.26 - 7.26i)T + 121iT^{2} \)
17 \( 1 - 18.9iT - 289T^{2} \)
19 \( 1 + (-23.3 + 23.3i)T - 361iT^{2} \)
23 \( 1 - 4.14iT - 529T^{2} \)
29 \( 1 + 17.3T + 841T^{2} \)
31 \( 1 + (22.1 - 22.1i)T - 961iT^{2} \)
37 \( 1 + (26.1 + 26.1i)T + 1.36e3iT^{2} \)
41 \( 1 + (-18.2 + 18.2i)T - 1.68e3iT^{2} \)
43 \( 1 + 15.3iT - 1.84e3T^{2} \)
47 \( 1 + (7.01 + 7.01i)T + 2.20e3iT^{2} \)
53 \( 1 + 61.6T + 2.80e3T^{2} \)
59 \( 1 + (-46.7 - 46.7i)T + 3.48e3iT^{2} \)
61 \( 1 + 7.30T + 3.72e3T^{2} \)
67 \( 1 + (-27.7 + 27.7i)T - 4.48e3iT^{2} \)
71 \( 1 + (-76.0 + 76.0i)T - 5.04e3iT^{2} \)
73 \( 1 + (-67.4 - 67.4i)T + 5.32e3iT^{2} \)
79 \( 1 - 11.8T + 6.24e3T^{2} \)
83 \( 1 + (111. - 111. i)T - 6.88e3iT^{2} \)
89 \( 1 + (-118. - 118. i)T + 7.92e3iT^{2} \)
97 \( 1 + (-109. + 109. i)T - 9.40e3iT^{2} \)
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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

−9.393414608842057570169273320115, −9.200463587745714455253801211364, −8.377589280969878679996396397663, −7.54618829281438767860894015815, −6.78352667945824882965298744762, −5.53350395874125104491748727058, −4.99883567721763212216075279823, −3.69375341744616813034194086961, −2.18549038504650819548938580336, −1.39412735922902937860612946055, 0.864677402408173691937137951705, 1.85599317022744716442156756986, 3.16562801958643665553331441587, 3.96940858172753218863432808233, 5.04517162102751305972312593530, 6.36119307299258339406625896920, 7.46561914949692339300952333662, 7.88289389549361044310793534613, 8.816568644369094218693912160856, 9.691871077410808111073767789515

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