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Properties

Label 2-1014-13.3-c1-0-24
Degree $2$
Conductor $1014$
Sign $-0.999 + 0.0256i$
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 − 2·5-s + (0.499 − 0.866i)6-s + (2 − 3.46i)7-s + 0.999·8-s + (−0.499 + 0.866i)9-s + (1 + 1.73i)10-s + (−2 − 3.46i)11-s − 0.999·12-s − 3.99·14-s + (−1 − 1.73i)15-s + (−0.5 − 0.866i)16-s + (−1 + 1.73i)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 − 0.894·5-s + (0.204 − 0.353i)6-s + (0.755 − 1.30i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (0.316 + 0.547i)10-s + (−0.603 − 1.04i)11-s − 0.288·12-s − 1.06·14-s + (−0.258 − 0.447i)15-s + (−0.125 − 0.216i)16-s + (−0.242 + 0.420i)17-s + 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1014\)    =    \(2 \cdot 3 \cdot 13^{2}\)
Sign: $-0.999 + 0.0256i$
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: \((2,\ 1014,\ (\ :1/2),\ -0.999 + 0.0256i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4022785464\)
\(L(\frac12)\) \(\approx\) \(0.4022785464\)
\(L(\frac{3}{2})\) 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 \( 1 + 2T + 5T^{2} \)
7 \( 1 + (-2 + 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5 + 8.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (8 + 13.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4 - 6.92i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.798364260814972238884862418113, −8.437538714039520507205622594830, −8.135884460639258980653108170615, −7.49152982329765896674352078523, −6.10774924336476300793289181373, −4.75662507632535100368520916332, −3.94690539412875462211810166483, −3.41033223829196181202537127968, −1.80490141972538580177281045757, −0.19250200304708857246001916242, 1.83989499757516797369505272854, 2.87820913814768552798286213066, 4.58148990842437095798915005268, 5.09106181565756656540423357970, 6.35877488863836362857121327480, 7.18880109604546421318558008814, 7.894324956602114837988826055505, 8.595900598507780154457311413929, 9.145368391691606469633274913217, 10.23293368755067370434529154658

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