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Properties

Label 1-1045-1045.538-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.887 + 0.460i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
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  + (0.984 + 0.173i)2-s + (−0.642 + 0.766i)3-s + (0.939 + 0.342i)4-s + (−0.766 + 0.642i)6-s + (0.866 − 0.5i)7-s + (0.866 + 0.5i)8-s + (−0.173 − 0.984i)9-s + (−0.866 + 0.5i)12-s + (−0.642 − 0.766i)13-s + (0.939 − 0.342i)14-s + (0.766 + 0.642i)16-s + (0.984 + 0.173i)17-s − i·18-s + (−0.173 + 0.984i)21-s + (0.342 − 0.939i)23-s + (−0.939 + 0.342i)24-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)2-s + (−0.642 + 0.766i)3-s + (0.939 + 0.342i)4-s + (−0.766 + 0.642i)6-s + (0.866 − 0.5i)7-s + (0.866 + 0.5i)8-s + (−0.173 − 0.984i)9-s + (−0.866 + 0.5i)12-s + (−0.642 − 0.766i)13-s + (0.939 − 0.342i)14-s + (0.766 + 0.642i)16-s + (0.984 + 0.173i)17-s − i·18-s + (−0.173 + 0.984i)21-s + (0.342 − 0.939i)23-s + (−0.939 + 0.342i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.887 + 0.460i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.887 + 0.460i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.588942692 + 0.6314591356i\)
\(L(\frac12)\) \(\approx\) \(2.588942692 + 0.6314591356i\)
\(L(1)\) \(\approx\) \(1.745382759 + 0.3953908316i\)
\(L(1)\) \(\approx\) \(1.745382759 + 0.3953908316i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.984 + 0.173i)T \)
3 \( 1 + (-0.642 + 0.766i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (-0.642 - 0.766i)T \)
17 \( 1 + (0.984 + 0.173i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.766 - 0.642i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (0.984 - 0.173i)T \)
53 \( 1 + (0.342 - 0.939i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (0.984 - 0.173i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.642 - 0.766i)T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (-0.984 - 0.173i)T \)
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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

−21.724215685492542873333187647844, −20.91752898788984845734859113991, −20.004288165435704145097625765176, −19.00447595017953887155091272145, −18.66092343681357006490476887180, −17.40887470340979726519431963181, −16.89999530758259859612478914399, −15.90252308737648190752089297372, −15.036534572009458977937992688043, −14.16540404364583822588675836723, −13.66737378819557059518625432323, −12.60185317563136409610086533584, −11.94131322481546696012366806029, −11.54317109812182219032661069556, −10.66318357287994438937694039280, −9.63649616369842346894345745924, −8.24107030300317095943842200478, −7.4175040066035652682511093529, −6.72283528403186157248760859670, −5.63903318220417823253463344457, −5.18304792580047059801268518130, −4.28064094712836361895387835099, −2.938062479706129818672685444776, −1.984201553543429717872593217393, −1.240144601943692271466244378702, 1.02120246807696004118227422802, 2.458566548431364093928594287525, 3.56950160760509250175467562566, 4.293277717936122903669071229690, 5.213109488651145270710291838151, 5.60116177944194040776998732590, 6.81000410881358791949650596815, 7.592148190907150972785718055927, 8.55884196943123605584085373804, 9.92904351854978310901935814675, 10.65522996900297589234846061384, 11.215233336119751298048004231634, 12.1972601622110158120277923011, 12.70155014986094706360147876658, 13.91778024841432556310245157991, 14.76091882370483234836278316321, 15.003858489506051788493248406314, 16.17031209784024551363664358558, 16.74574185968192101173869101619, 17.39292708827999567625409789919, 18.24758208240955463863414751939, 19.61329410318770927075637972012, 20.500550487573117356127238350651, 20.89307828317689909901034353782, 21.70135746994231357274909990628

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