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Properties

Label 2-1029-1029.365-c0-0-0
Degree $2$
Conductor $1029$
Sign $-0.118 - 0.992i$
Analytic cond. $0.513537$
Root an. cond. $0.716615$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.404 + 0.914i)3-s + (−0.0960 + 0.995i)4-s + (0.926 + 0.375i)7-s + (−0.672 + 0.740i)9-s + (−0.949 + 0.315i)12-s + (0.529 − 0.758i)13-s + (−0.981 − 0.191i)16-s + 0.319·19-s + (0.0320 + 0.999i)21-s + (−0.572 − 0.820i)25-s + (−0.949 − 0.315i)27-s + (−0.462 + 0.886i)28-s + (0.173 − 0.0833i)31-s + (−0.672 − 0.740i)36-s + (−1.66 − 1.08i)37-s + ⋯
L(s)  = 1  + (0.404 + 0.914i)3-s + (−0.0960 + 0.995i)4-s + (0.926 + 0.375i)7-s + (−0.672 + 0.740i)9-s + (−0.949 + 0.315i)12-s + (0.529 − 0.758i)13-s + (−0.981 − 0.191i)16-s + 0.319·19-s + (0.0320 + 0.999i)21-s + (−0.572 − 0.820i)25-s + (−0.949 − 0.315i)27-s + (−0.462 + 0.886i)28-s + (0.173 − 0.0833i)31-s + (−0.672 − 0.740i)36-s + (−1.66 − 1.08i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1029\)    =    \(3 \cdot 7^{3}\)
Sign: $-0.118 - 0.992i$
Analytic conductor: \(0.513537\)
Root analytic conductor: \(0.716615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1029} (365, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1029,\ (\ :0),\ -0.118 - 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.211826965\)
\(L(\frac12)\) \(\approx\) \(1.211826965\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.404 - 0.914i)T \)
7 \( 1 + (-0.926 - 0.375i)T \)
good2 \( 1 + (0.0960 - 0.995i)T^{2} \)
5 \( 1 + (0.572 + 0.820i)T^{2} \)
11 \( 1 + (0.838 - 0.545i)T^{2} \)
13 \( 1 + (-0.529 + 0.758i)T + (-0.345 - 0.938i)T^{2} \)
17 \( 1 + (-0.404 - 0.914i)T^{2} \)
19 \( 1 - 0.319T + T^{2} \)
23 \( 1 + (-0.0320 - 0.999i)T^{2} \)
29 \( 1 + (-0.0320 + 0.999i)T^{2} \)
31 \( 1 + (-0.173 + 0.0833i)T + (0.623 - 0.781i)T^{2} \)
37 \( 1 + (1.66 + 1.08i)T + (0.404 + 0.914i)T^{2} \)
41 \( 1 + (-0.159 - 0.987i)T^{2} \)
43 \( 1 + (-0.456 - 0.340i)T + (0.284 + 0.958i)T^{2} \)
47 \( 1 + (-0.991 + 0.127i)T^{2} \)
53 \( 1 + (0.462 + 0.886i)T^{2} \)
59 \( 1 + (-0.159 + 0.987i)T^{2} \)
61 \( 1 + (0.358 - 0.972i)T + (-0.761 - 0.648i)T^{2} \)
67 \( 1 + (0.729 + 0.351i)T + (0.623 + 0.781i)T^{2} \)
71 \( 1 + (-0.0320 - 0.999i)T^{2} \)
73 \( 1 + (-1.67 - 0.107i)T + (0.991 + 0.127i)T^{2} \)
79 \( 1 + (-0.153 + 0.673i)T + (-0.900 - 0.433i)T^{2} \)
83 \( 1 + (-0.718 + 0.695i)T^{2} \)
89 \( 1 + (-0.518 - 0.855i)T^{2} \)
97 \( 1 + (1.57 - 0.756i)T + (0.623 - 0.781i)T^{2} \)
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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

−10.42614219755057298115817069072, −9.357092257521423910829403258834, −8.617170767574404481037460448220, −8.103632746376123702954638047792, −7.36249478112840317405519142140, −5.89361753792337811032145770654, −4.98749916408183512660240826733, −4.11506781718504565728373394849, −3.25723196962773468292572497370, −2.21688100671538990608519803328, 1.27029033419457962316436636992, 2.03488107001003386254214594457, 3.60265896774173846210843839615, 4.80113352142138925367843949246, 5.70366726311321171403408245241, 6.63040236834292146821754346137, 7.32869652624989682326183233822, 8.306375310535527202367162744204, 8.991770849055219378000096339734, 9.838705056688771563968948291360

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