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Properties

Label 1-1012-1012.303-r0-0-0
Degree $1$
Conductor $1012$
Sign $-0.284 - 0.958i$
Analytic cond. $4.69970$
Root an. cond. $4.69970$
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  + (−0.774 − 0.633i)3-s + (0.198 − 0.980i)5-s + (−0.0285 + 0.999i)7-s + (0.198 + 0.980i)9-s + (−0.941 + 0.336i)13-s + (−0.774 + 0.633i)15-s + (−0.696 + 0.717i)17-s + (0.897 − 0.441i)19-s + (0.654 − 0.755i)21-s + (−0.921 − 0.389i)25-s + (0.466 − 0.884i)27-s + (−0.897 − 0.441i)29-s + (0.998 − 0.0570i)31-s + (0.974 + 0.226i)35-s + (−0.736 − 0.676i)37-s + ⋯
L(s)  = 1  + (−0.774 − 0.633i)3-s + (0.198 − 0.980i)5-s + (−0.0285 + 0.999i)7-s + (0.198 + 0.980i)9-s + (−0.941 + 0.336i)13-s + (−0.774 + 0.633i)15-s + (−0.696 + 0.717i)17-s + (0.897 − 0.441i)19-s + (0.654 − 0.755i)21-s + (−0.921 − 0.389i)25-s + (0.466 − 0.884i)27-s + (−0.897 − 0.441i)29-s + (0.998 − 0.0570i)31-s + (0.974 + 0.226i)35-s + (−0.736 − 0.676i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.284 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $-0.284 - 0.958i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ -0.284 - 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4927680667 - 0.6603741127i\)
\(L(\frac12)\) \(\approx\) \(0.4927680667 - 0.6603741127i\)
\(L(1)\) \(\approx\) \(0.7384775787 - 0.2458973792i\)
\(L(1)\) \(\approx\) \(0.7384775787 - 0.2458973792i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (-0.774 - 0.633i)T \)
5 \( 1 + (0.198 - 0.980i)T \)
7 \( 1 + (-0.0285 + 0.999i)T \)
13 \( 1 + (-0.941 + 0.336i)T \)
17 \( 1 + (-0.696 + 0.717i)T \)
19 \( 1 + (0.897 - 0.441i)T \)
29 \( 1 + (-0.897 - 0.441i)T \)
31 \( 1 + (0.998 - 0.0570i)T \)
37 \( 1 + (-0.736 - 0.676i)T \)
41 \( 1 + (0.736 - 0.676i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.610 - 0.791i)T \)
59 \( 1 + (0.564 - 0.825i)T \)
61 \( 1 + (0.362 + 0.931i)T \)
67 \( 1 + (0.654 - 0.755i)T \)
71 \( 1 + (-0.974 + 0.226i)T \)
73 \( 1 + (0.985 + 0.170i)T \)
79 \( 1 + (0.941 - 0.336i)T \)
83 \( 1 + (-0.870 + 0.491i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (-0.870 - 0.491i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−22.14870800368856320971308278370, −21.121589918240612411144712871, −20.41448289566744098999322433552, −19.58834863820552840214860350032, −18.561980691111209260397306189070, −17.7320509610200077646080921647, −17.30560686544689232309489806655, −16.36757498982475724082685454449, −15.65307130022836406795825955193, −14.74975020302630807822252820535, −14.107255510763389955418585540848, −13.19339960614435478514238794780, −12.077932720555251777156509118736, −11.294465453566401225834691167, −10.64256844904579711938396809081, −9.92039894244502371455523822174, −9.37133685029270259723203960008, −7.753218506313882265717676994255, −7.07599021307757307501020490287, −6.32968425440702470316039566330, −5.310328973480881315217535740804, −4.46377625222918149195917486389, −3.51726523604083252289052253931, −2.62900458866409393659628221679, −1.04705034705145473476876937109, 0.448522343564000824281935055759, 1.80742594765143458121416942962, 2.41160423732331457540367736697, 4.11098779421034490625999897869, 5.14605173659436856376339462767, 5.56499075101585150867631357879, 6.547714223170607917238311621561, 7.49011095358072583649513497900, 8.44357453275694166793715511424, 9.21493042394357242051520136361, 10.08192867804481229156715897747, 11.29297419974456221301669865863, 11.94788679235966485185481555848, 12.55159926749657080207571408845, 13.21787842567952872292101889200, 14.09796078213089860898140826604, 15.31051207919394085628853317189, 15.98012298651148839259466363384, 16.84132188035009166318004177591, 17.51553623985749367682501046350, 18.063665103146534381348529332833, 19.20093561665610586490511704972, 19.5348563556788076971651409584, 20.704722287972975081831903895507, 21.5450293454602773783213188728

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