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Properties

Label 2-1029-1029.407-c0-0-0
Degree $2$
Conductor $1029$
Sign $0.947 - 0.319i$
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  + (0.991 − 0.127i)3-s + (0.871 + 0.490i)4-s + (−0.462 + 0.886i)7-s + (0.967 − 0.253i)9-s + (0.926 + 0.375i)12-s + (−0.678 − 1.53i)13-s + (0.518 + 0.855i)16-s − 1.96·19-s + (−0.345 + 0.938i)21-s + (0.404 − 0.914i)25-s + (0.926 − 0.375i)27-s + (−0.838 + 0.545i)28-s + (−0.387 + 1.69i)31-s + (0.967 + 0.253i)36-s + (−0.318 + 0.0204i)37-s + ⋯
L(s)  = 1  + (0.991 − 0.127i)3-s + (0.871 + 0.490i)4-s + (−0.462 + 0.886i)7-s + (0.967 − 0.253i)9-s + (0.926 + 0.375i)12-s + (−0.678 − 1.53i)13-s + (0.518 + 0.855i)16-s − 1.96·19-s + (−0.345 + 0.938i)21-s + (0.404 − 0.914i)25-s + (0.926 − 0.375i)27-s + (−0.838 + 0.545i)28-s + (−0.387 + 1.69i)31-s + (0.967 + 0.253i)36-s + (−0.318 + 0.0204i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1029\)    =    \(3 \cdot 7^{3}\)
Sign: $0.947 - 0.319i$
Analytic conductor: \(0.513537\)
Root analytic conductor: \(0.716615\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1029} (407, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1029,\ (\ :0),\ 0.947 - 0.319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.571205487\)
\(L(\frac12)\) \(\approx\) \(1.571205487\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.991 + 0.127i)T \)
7 \( 1 + (0.462 - 0.886i)T \)
good2 \( 1 + (-0.871 - 0.490i)T^{2} \)
5 \( 1 + (-0.404 + 0.914i)T^{2} \)
11 \( 1 + (0.997 + 0.0640i)T^{2} \)
13 \( 1 + (0.678 + 1.53i)T + (-0.672 + 0.740i)T^{2} \)
17 \( 1 + (-0.991 + 0.127i)T^{2} \)
19 \( 1 + 1.96T + T^{2} \)
23 \( 1 + (0.345 - 0.938i)T^{2} \)
29 \( 1 + (0.345 + 0.938i)T^{2} \)
31 \( 1 + (0.387 - 1.69i)T + (-0.900 - 0.433i)T^{2} \)
37 \( 1 + (0.318 - 0.0204i)T + (0.991 - 0.127i)T^{2} \)
41 \( 1 + (0.981 + 0.191i)T^{2} \)
43 \( 1 + (-0.0460 + 0.0445i)T + (0.0320 - 0.999i)T^{2} \)
47 \( 1 + (-0.159 - 0.987i)T^{2} \)
53 \( 1 + (0.838 + 0.545i)T^{2} \)
59 \( 1 + (0.981 - 0.191i)T^{2} \)
61 \( 1 + (0.382 + 0.421i)T + (-0.0960 + 0.995i)T^{2} \)
67 \( 1 + (0.441 + 1.93i)T + (-0.900 + 0.433i)T^{2} \)
71 \( 1 + (0.345 - 0.938i)T^{2} \)
73 \( 1 + (-1.51 + 1.29i)T + (0.159 - 0.987i)T^{2} \)
79 \( 1 + (0.838 - 1.05i)T + (-0.222 - 0.974i)T^{2} \)
83 \( 1 + (0.572 - 0.820i)T^{2} \)
89 \( 1 + (-0.284 - 0.958i)T^{2} \)
97 \( 1 + (0.356 - 1.56i)T + (-0.900 - 0.433i)T^{2} \)
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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

−10.33016195714492256420320356548, −9.153032500951053119146975549028, −8.398084351975119087393417737972, −7.889549520183340838136019249204, −6.83968225465693590779545057593, −6.21653835319105097637513572375, −4.93005598966655705763942087624, −3.55268908777920400166271146055, −2.79015943428700923209479444921, −2.05183761321917210076576245250, 1.72611657687410586405059253153, 2.57610285952999954784985384284, 3.86747159458894856949827795827, 4.57740148448832279134837701782, 6.08468895860829206209560313781, 6.99293974978080426062823387661, 7.32434869498527262614011614884, 8.502435274124111496313529050068, 9.419066360033732395563288923480, 9.992571670900321601363101173289

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