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Properties

Label 1-1012-1012.127-r0-0-0
Degree $1$
Conductor $1012$
Sign $0.524 - 0.851i$
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  + (0.0285 + 0.999i)3-s + (−0.998 − 0.0570i)5-s + (0.897 + 0.441i)7-s + (−0.998 + 0.0570i)9-s + (−0.696 − 0.717i)13-s + (0.0285 − 0.999i)15-s + (−0.974 + 0.226i)17-s + (0.516 − 0.856i)19-s + (−0.415 + 0.909i)21-s + (0.993 + 0.113i)25-s + (−0.0855 − 0.996i)27-s + (−0.516 − 0.856i)29-s + (−0.610 + 0.791i)31-s + (−0.870 − 0.491i)35-s + (0.774 − 0.633i)37-s + ⋯
L(s)  = 1  + (0.0285 + 0.999i)3-s + (−0.998 − 0.0570i)5-s + (0.897 + 0.441i)7-s + (−0.998 + 0.0570i)9-s + (−0.696 − 0.717i)13-s + (0.0285 − 0.999i)15-s + (−0.974 + 0.226i)17-s + (0.516 − 0.856i)19-s + (−0.415 + 0.909i)21-s + (0.993 + 0.113i)25-s + (−0.0855 − 0.996i)27-s + (−0.516 − 0.856i)29-s + (−0.610 + 0.791i)31-s + (−0.870 − 0.491i)35-s + (0.774 − 0.633i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.524 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $0.524 - 0.851i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ 0.524 - 0.851i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5679528601 - 0.3169799643i\)
\(L(\frac12)\) \(\approx\) \(0.5679528601 - 0.3169799643i\)
\(L(1)\) \(\approx\) \(0.7775784990 + 0.1432817708i\)
\(L(1)\) \(\approx\) \(0.7775784990 + 0.1432817708i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.0285 + 0.999i)T \)
5 \( 1 + (-0.998 - 0.0570i)T \)
7 \( 1 + (0.897 + 0.441i)T \)
13 \( 1 + (-0.696 - 0.717i)T \)
17 \( 1 + (-0.974 + 0.226i)T \)
19 \( 1 + (0.516 - 0.856i)T \)
29 \( 1 + (-0.516 - 0.856i)T \)
31 \( 1 + (-0.610 + 0.791i)T \)
37 \( 1 + (0.774 - 0.633i)T \)
41 \( 1 + (-0.774 - 0.633i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.466 - 0.884i)T \)
59 \( 1 + (0.985 + 0.170i)T \)
61 \( 1 + (-0.941 - 0.336i)T \)
67 \( 1 + (-0.415 + 0.909i)T \)
71 \( 1 + (0.870 - 0.491i)T \)
73 \( 1 + (0.921 - 0.389i)T \)
79 \( 1 + (0.696 + 0.717i)T \)
83 \( 1 + (-0.362 - 0.931i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (-0.362 + 0.931i)T \)
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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

−21.920087357257609197920761381223, −20.63687284828148599676069551041, −20.0910764875660211167962084664, −19.48326678171336774189183540190, −18.45186135720143004242907627193, −18.18131155677162779266663773395, −16.9502140407552693494301600039, −16.55561451530043355758644700502, −15.23967196880786461863391050649, −14.57955646919588171576725784822, −13.89822490876043963498200586824, −12.961536810111617356775890830332, −12.10076171728696405664283703429, −11.44182525879911197857351636605, −10.94690139874867054821137500974, −9.54240892859831779509103542614, −8.50296254036938617816119894622, −7.807616450730756187494906418007, −7.23253057613273362944043935224, −6.43552374900371638817894093990, −5.13602411721651583976466364097, −4.33805993952917294983537750919, −3.26466802055455315859176926832, −2.09968184344243366742722075925, −1.18255579221085139328174930805, 0.29271243520177475455480721222, 2.12897951333590969256524476172, 3.14205351555762213489558669140, 4.0796574578943914814687667676, 4.90722381932839035944324129038, 5.43923430895064086264328170260, 6.879594641741996444143061508415, 7.88325520393283553431746100769, 8.54797744166340974431137975836, 9.30125672444971718608080598023, 10.36956761941209898952628128077, 11.2065884896887437434682742228, 11.62823291221986317824309896600, 12.59207346187575684882408485517, 13.72392767459402077752287794238, 14.83445012845388128918522649133, 15.15554232093802290497161685, 15.790290654142297260446559757053, 16.706864724951630734471747712662, 17.54996793873510733993545251834, 18.24302383128845961364725814031, 19.47221063631208863077485034006, 20.014959567019419187416449603729, 20.62564850561120725392097404116, 21.62083016688548998429482653293

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