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Properties

Label 2-1014-13.8-c2-0-25
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 + (−1.26 − 1.26i)5-s + (−1.73 − 1.73i)6-s + (−0.732 + 0.732i)7-s + (2 − 2i)8-s + 2.99·9-s + 2.53i·10-s + (1.73 − 1.73i)11-s + 3.46i·12-s + 1.46·14-s + (−2.19 − 2.19i)15-s − 4·16-s + 5.32i·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 + (−0.104 + 0.104i)7-s + (0.250 − 0.250i)8-s + 0.333·9-s + 0.253i·10-s + (0.157 − 0.157i)11-s + 0.288i·12-s + 0.104·14-s + (−0.146 − 0.146i)15-s − 0.250·16-s + 0.312i·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

\(L(\frac{3}{2})\) \(\approx\) \(1.732377305\)
\(L(\frac12)\) \(\approx\) \(1.732377305\)
\(L(2)\) 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 \( 1 + (0.732 - 0.732i)T - 49iT^{2} \)
11 \( 1 + (-1.73 + 1.73i)T - 121iT^{2} \)
17 \( 1 - 5.32iT - 289T^{2} \)
19 \( 1 + (-14.7 - 14.7i)T + 361iT^{2} \)
23 \( 1 - 5.32iT - 529T^{2} \)
29 \( 1 - 4.14T + 841T^{2} \)
31 \( 1 + (-24.9 - 24.9i)T + 961iT^{2} \)
37 \( 1 + (-3.14 + 3.14i)T - 1.36e3iT^{2} \)
41 \( 1 + (44.4 + 44.4i)T + 1.68e3iT^{2} \)
43 \( 1 - 37.1iT - 1.84e3T^{2} \)
47 \( 1 + (-30.8 + 30.8i)T - 2.20e3iT^{2} \)
53 \( 1 - 57.7T + 2.80e3T^{2} \)
59 \( 1 + (-66.6 + 66.6i)T - 3.48e3iT^{2} \)
61 \( 1 - 103.T + 3.72e3T^{2} \)
67 \( 1 + (46.6 + 46.6i)T + 4.48e3iT^{2} \)
71 \( 1 + (-26.9 - 26.9i)T + 5.04e3iT^{2} \)
73 \( 1 + (5.67 - 5.67i)T - 5.32e3iT^{2} \)
79 \( 1 - 4.21T + 6.24e3T^{2} \)
83 \( 1 + (-109. - 109. i)T + 6.88e3iT^{2} \)
89 \( 1 + (-19.5 + 19.5i)T - 7.92e3iT^{2} \)
97 \( 1 + (4.03 + 4.03i)T + 9.40e3iT^{2} \)
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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

−9.792445069521719974308679101876, −8.752999773675496136360026164464, −8.309270934456890559802992481363, −7.44144234130706364986046897964, −6.48430804795055567557933416461, −5.25517521556772121726288937409, −4.07502518039284188908941231113, −3.27879807225990424080896429377, −2.14149380697994746117407630618, −0.894516102222016927445238338430, 0.854215500561437292741241704593, 2.35074224672804575744251775613, 3.46126753111715763489982446503, 4.59905220651031598906255583930, 5.61560138042212646409016162531, 6.81316542095079358634302932126, 7.27951814729233213686188588627, 8.183917867430231982334407481602, 8.938813923952307414927257111965, 9.701402973726662467978850596611

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