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Properties

Label 1-1045-1045.339-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.782 + 0.622i$
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  + (−0.438 + 0.898i)2-s + (0.374 + 0.927i)3-s + (−0.615 − 0.788i)4-s + (−0.997 − 0.0697i)6-s + (0.978 + 0.207i)7-s + (0.978 − 0.207i)8-s + (−0.719 + 0.694i)9-s + (0.5 − 0.866i)12-s + (0.241 + 0.970i)13-s + (−0.615 + 0.788i)14-s + (−0.241 + 0.970i)16-s + (0.719 + 0.694i)17-s + (−0.309 − 0.951i)18-s + (0.173 + 0.984i)21-s + (0.939 − 0.342i)23-s + (0.559 + 0.829i)24-s + ⋯
L(s)  = 1  + (−0.438 + 0.898i)2-s + (0.374 + 0.927i)3-s + (−0.615 − 0.788i)4-s + (−0.997 − 0.0697i)6-s + (0.978 + 0.207i)7-s + (0.978 − 0.207i)8-s + (−0.719 + 0.694i)9-s + (0.5 − 0.866i)12-s + (0.241 + 0.970i)13-s + (−0.615 + 0.788i)14-s + (−0.241 + 0.970i)16-s + (0.719 + 0.694i)17-s + (−0.309 − 0.951i)18-s + (0.173 + 0.984i)21-s + (0.939 − 0.342i)23-s + (0.559 + 0.829i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.782 + 0.622i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.782 + 0.622i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5020540863 + 1.437260415i\)
\(L(\frac12)\) \(\approx\) \(0.5020540863 + 1.437260415i\)
\(L(1)\) \(\approx\) \(0.7692792611 + 0.7674802862i\)
\(L(1)\) \(\approx\) \(0.7692792611 + 0.7674802862i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.438 + 0.898i)T \)
3 \( 1 + (0.374 + 0.927i)T \)
7 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (0.241 + 0.970i)T \)
17 \( 1 + (0.719 + 0.694i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.990 - 0.139i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-0.374 - 0.927i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (0.882 + 0.469i)T \)
53 \( 1 + (-0.961 - 0.275i)T \)
59 \( 1 + (-0.882 + 0.469i)T \)
61 \( 1 + (0.559 - 0.829i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (0.961 - 0.275i)T \)
73 \( 1 + (-0.848 - 0.529i)T \)
79 \( 1 + (-0.997 + 0.0697i)T \)
83 \( 1 + (0.104 + 0.994i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (-0.438 + 0.898i)T \)
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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

−20.892149974077660519213856444489, −20.43325414158259783687564023914, −19.72491766191272147937960076260, −18.81427114759811673963955806522, −18.33777007723796057833482715826, −17.44633552244111089211321091652, −17.15840663271122182137874585394, −15.75063369113996331908315080231, −14.65192906746352895130668353018, −13.87537686479820589754185024194, −13.29683977514359292734755493762, −12.33317213520035786471921390879, −11.7841392705417631090366767872, −10.93741914604174664589381252669, −10.10806571469951714303850072294, −9.04473451026448704413711627185, −8.24926675493651196045094623996, −7.74805812855188650061192773979, −6.87515285397249418337646217971, −5.48118219847751996487144356312, −4.57766538605774856499849312232, −3.24306013635595276811726011724, −2.70746081142940944138336150093, −1.44851636129213574299787225842, −0.88164670212556314573218897201, 1.25450694734100319043569341955, 2.435368526148383689798122092517, 3.91552118881095869085020982045, 4.602305122602681741471155944997, 5.38622804764235828824259611109, 6.28162019361679414151255709688, 7.44203556622502237490892281193, 8.26521930260823718904742224188, 8.84548445303114934498445828283, 9.59811012934944603821657557868, 10.568092923982257151530583863525, 11.122691151065915923809413400776, 12.29890228666652350886425437207, 13.70558620586709834417050222226, 14.24279196774416099301122357602, 14.8534004550579430442467554250, 15.60387819475640014614309722426, 16.29356240155346695799878677025, 17.16402605303620299553786981241, 17.59820588530243831527654636684, 18.9381976293806744724076023578, 19.15505962440752573157568681189, 20.40191747888186631557083536500, 21.134520127402839483032307180338, 21.72007047610908427560234674663

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