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Properties

Label 2-1014-13.3-c1-0-17
Degree $2$
Conductor $1014$
Sign $0.755 + 0.655i$
Analytic cond. $8.09683$
Root an. cond. $2.84549$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + 1.73·5-s + (0.499 − 0.866i)6-s + (2.36 − 4.09i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.866 − 1.49i)10-s + (2.36 + 4.09i)11-s − 0.999·12-s − 4.73·14-s + (0.866 + 1.49i)15-s + (−0.5 − 0.866i)16-s + (2.59 − 4.5i)17-s + 0.999·18-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + 0.774·5-s + (0.204 − 0.353i)6-s + (0.894 − 1.54i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.273 − 0.474i)10-s + (0.713 + 1.23i)11-s − 0.288·12-s − 1.26·14-s + (0.223 + 0.387i)15-s + (−0.125 − 0.216i)16-s + (0.630 − 1.09i)17-s + 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $0.755 + 0.655i$
Analytic conductor: \(8.09683\)
Root analytic conductor: \(2.84549\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1014} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1014,\ (\ :1/2),\ 0.755 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.897738488\)
\(L(\frac12)\) \(\approx\) \(1.897738488\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 \)
good5 \( 1 - 1.73T + 5T^{2} \)
7 \( 1 + (-2.36 + 4.09i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.36 - 4.09i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.59 + 4.5i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.633 + 1.09i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.09 + 1.90i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.53T + 31T^{2} \)
37 \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.232 + 0.401i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.09 - 5.36i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 1.26T + 47T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (-6.92 + 12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.40 - 4.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.36 - 9.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.09 + 7.09i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 - 11.6T + 83T^{2} \)
89 \( 1 + (-1.26 - 2.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3 + 5.19i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.802388415261785615420708096046, −9.471454119315831046714379225232, −8.280499080235791499732860532041, −7.45773483788176518416016642276, −6.75598840639314417028290807737, −5.14382556717242297272576842049, −4.47653124226387003591104945691, −3.59447505604814877523106573955, −2.20124469829245014992331495060, −1.15583859476880636239869723132, 1.40079287153463705573707305331, 2.31489505239269390877729029464, 3.75376449236174848968684612391, 5.41389751332613500273024066505, 5.78697512592380010045927215883, 6.48239898712289480976095206789, 7.82338168817410491683973240381, 8.396301813884124602098685243257, 8.987690496337804351557009587729, 9.719822693442540145805670706402

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