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Properties

Label 2-1014-13.8-c2-0-40
Degree $2$
Conductor $1014$
Sign $-0.881 - 0.471i$
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 + (2.73 − 2.73i)7-s + (2 − 2i)8-s + 2.99·9-s + 9.46i·10-s + (−1.73 + 1.73i)11-s − 3.46i·12-s − 5.46·14-s + (8.19 + 8.19i)15-s − 4·16-s − 29.3i·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 + (0.390 − 0.390i)7-s + (0.250 − 0.250i)8-s + 0.333·9-s + 0.946i·10-s + (−0.157 + 0.157i)11-s − 0.288i·12-s − 0.390·14-s + (0.546 + 0.546i)15-s − 0.250·16-s − 1.72i·17-s + (−0.166 − 0.166i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.881 - 0.471i$
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: \((2,\ 1014,\ (\ :1),\ -0.881 - 0.471i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5448246471\)
\(L(\frac12)\) \(\approx\) \(0.5448246471\)
\(L(2)\) 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 \( 1 + (-2.73 + 2.73i)T - 49iT^{2} \)
11 \( 1 + (1.73 - 1.73i)T - 121iT^{2} \)
17 \( 1 + 29.3iT - 289T^{2} \)
19 \( 1 + (-11.2 - 11.2i)T + 361iT^{2} \)
23 \( 1 + 29.3iT - 529T^{2} \)
29 \( 1 - 31.8T + 841T^{2} \)
31 \( 1 + (26.9 + 26.9i)T + 961iT^{2} \)
37 \( 1 + (-30.8 + 30.8i)T - 1.36e3iT^{2} \)
41 \( 1 + (-14.4 - 14.4i)T + 1.68e3iT^{2} \)
43 \( 1 + 25.1iT - 1.84e3T^{2} \)
47 \( 1 + (-41.1 + 41.1i)T - 2.20e3iT^{2} \)
53 \( 1 - 2.28T + 2.80e3T^{2} \)
59 \( 1 + (54.6 - 54.6i)T - 3.48e3iT^{2} \)
61 \( 1 + 7.42T + 3.72e3T^{2} \)
67 \( 1 + (-60.6 - 60.6i)T + 4.48e3iT^{2} \)
71 \( 1 + (38.9 + 38.9i)T + 5.04e3iT^{2} \)
73 \( 1 + (40.3 - 40.3i)T - 5.32e3iT^{2} \)
79 \( 1 + 148.T + 6.24e3T^{2} \)
83 \( 1 + (73.7 + 73.7i)T + 6.88e3iT^{2} \)
89 \( 1 + (25.5 - 25.5i)T - 7.92e3iT^{2} \)
97 \( 1 + (-86.0 - 86.0i)T + 9.40e3iT^{2} \)
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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

−9.297225934791044013549296651458, −8.504091089640581633319247011895, −7.64525280954539291124909140253, −7.11846999631521134477721575102, −5.66732307774343196320616465628, −4.63213910801004976328219874811, −4.12822541114516897500728299497, −2.67149833920406440975749280577, −1.06605481605288875036560337287, −0.27872729083299149190551868856, 1.41929881342642708982597681322, 3.02086074172713140816526472065, 4.13007319425211358611338347338, 5.26562633322573062815957993919, 6.13691273627329897032358084516, 6.94994195876238352412047627420, 7.72600080173559658313780513766, 8.349594366865984166048370532542, 9.387083589608401242234440600336, 10.39743496852278506626448720410

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