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Properties

Label 2-1014-13.8-c2-0-13
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 + (−4.41 − 4.41i)5-s + (1.73 + 1.73i)6-s + (−5.77 + 5.77i)7-s + (−2 + 2i)8-s + 2.99·9-s − 8.82i·10-s + (13.5 − 13.5i)11-s + 3.46i·12-s − 11.5·14-s + (−7.64 − 7.64i)15-s − 4·16-s + 23.0i·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 + (−0.882 − 0.882i)5-s + (0.288 + 0.288i)6-s + (−0.825 + 0.825i)7-s + (−0.250 + 0.250i)8-s + 0.333·9-s − 0.882i·10-s + (1.23 − 1.23i)11-s + 0.288i·12-s − 0.825·14-s + (−0.509 − 0.509i)15-s − 0.250·16-s + 1.35i·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: $\chi_{1014} (775, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1),\ -0.208 - 0.977i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.102116940\)
\(L(\frac12)\) \(\approx\) \(2.102116940\)
\(L(2)\) 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 \( 1 + (4.41 + 4.41i)T + 25iT^{2} \)
7 \( 1 + (5.77 - 5.77i)T - 49iT^{2} \)
11 \( 1 + (-13.5 + 13.5i)T - 121iT^{2} \)
17 \( 1 - 23.0iT - 289T^{2} \)
19 \( 1 + (-0.305 - 0.305i)T + 361iT^{2} \)
23 \( 1 - 42.1iT - 529T^{2} \)
29 \( 1 - 6.51T + 841T^{2} \)
31 \( 1 + (-17.8 - 17.8i)T + 961iT^{2} \)
37 \( 1 + (1.01 - 1.01i)T - 1.36e3iT^{2} \)
41 \( 1 + (-29.5 - 29.5i)T + 1.68e3iT^{2} \)
43 \( 1 - 59.0iT - 1.84e3T^{2} \)
47 \( 1 + (15.0 - 15.0i)T - 2.20e3iT^{2} \)
53 \( 1 - 8.90T + 2.80e3T^{2} \)
59 \( 1 + (31.1 - 31.1i)T - 3.48e3iT^{2} \)
61 \( 1 + 89.6T + 3.72e3T^{2} \)
67 \( 1 + (28.0 + 28.0i)T + 4.48e3iT^{2} \)
71 \( 1 + (-6.27 - 6.27i)T + 5.04e3iT^{2} \)
73 \( 1 + (-5.92 + 5.92i)T - 5.32e3iT^{2} \)
79 \( 1 - 115.T + 6.24e3T^{2} \)
83 \( 1 + (-34.2 - 34.2i)T + 6.88e3iT^{2} \)
89 \( 1 + (-42.5 + 42.5i)T - 7.92e3iT^{2} \)
97 \( 1 + (94.5 + 94.5i)T + 9.40e3iT^{2} \)
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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

−9.554319711447794486550453608905, −8.983391436514934795194091133757, −8.336656271579988540490919689359, −7.68264364029580320110724454628, −6.35965962660868262961720670639, −5.91506680403593706578153752421, −4.62788990614762703774912446915, −3.70196677002572146924464842222, −3.12654457106438234719432868712, −1.29939838527822643140746740837, 0.55946978424572345367654206928, 2.28229043088390759545423249054, 3.24720837056617210528480966251, 4.02318560853954895734545260120, 4.66054203550320143544773947188, 6.45288348010420714219218874415, 6.96801524179131136584890801726, 7.56583515039416035011376225905, 8.912568343375154133187195589982, 9.702533937785298954930049323146

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