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Properties

Label 1-1045-1045.257-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.910 + 0.413i$
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  + (−0.788 + 0.615i)2-s + (−0.694 − 0.719i)3-s + (0.241 − 0.970i)4-s + (0.990 + 0.139i)6-s + (−0.406 + 0.913i)7-s + (0.406 + 0.913i)8-s + (−0.0348 + 0.999i)9-s + (−0.866 + 0.5i)12-s + (−0.469 − 0.882i)13-s + (−0.241 − 0.970i)14-s + (−0.882 − 0.469i)16-s + (−0.999 + 0.0348i)17-s + (−0.587 − 0.809i)18-s + (0.939 − 0.342i)21-s + (−0.642 − 0.766i)23-s + (0.374 − 0.927i)24-s + ⋯
L(s)  = 1  + (−0.788 + 0.615i)2-s + (−0.694 − 0.719i)3-s + (0.241 − 0.970i)4-s + (0.990 + 0.139i)6-s + (−0.406 + 0.913i)7-s + (0.406 + 0.913i)8-s + (−0.0348 + 0.999i)9-s + (−0.866 + 0.5i)12-s + (−0.469 − 0.882i)13-s + (−0.241 − 0.970i)14-s + (−0.882 − 0.469i)16-s + (−0.999 + 0.0348i)17-s + (−0.587 − 0.809i)18-s + (0.939 − 0.342i)21-s + (−0.642 − 0.766i)23-s + (0.374 − 0.927i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.413i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.910 + 0.413i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.910 + 0.413i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5138244281 + 0.1111256910i\)
\(L(\frac12)\) \(\approx\) \(0.5138244281 + 0.1111256910i\)
\(L(1)\) \(\approx\) \(0.5153607799 + 0.05070188743i\)
\(L(1)\) \(\approx\) \(0.5153607799 + 0.05070188743i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.788 + 0.615i)T \)
3 \( 1 + (-0.694 - 0.719i)T \)
7 \( 1 + (-0.406 + 0.913i)T \)
13 \( 1 + (-0.469 - 0.882i)T \)
17 \( 1 + (-0.999 + 0.0348i)T \)
23 \( 1 + (-0.642 - 0.766i)T \)
29 \( 1 + (0.961 - 0.275i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (0.719 - 0.694i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (-0.829 + 0.559i)T \)
53 \( 1 + (-0.529 + 0.848i)T \)
59 \( 1 + (0.559 - 0.829i)T \)
61 \( 1 + (-0.374 - 0.927i)T \)
67 \( 1 + (-0.342 + 0.939i)T \)
71 \( 1 + (-0.848 + 0.529i)T \)
73 \( 1 + (0.898 - 0.438i)T \)
79 \( 1 + (0.990 - 0.139i)T \)
83 \( 1 + (0.207 + 0.978i)T \)
89 \( 1 + (0.173 + 0.984i)T \)
97 \( 1 + (-0.788 + 0.615i)T \)
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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

−21.42521345867562458658013710772, −20.73862634443063731534615529242, −19.729933719754257298075295320484, −19.50595201631891316107016673552, −18.03803434407774904102635155528, −17.75946464199309102271146307943, −16.69186856419585638313999686957, −16.405372602170952748177882068273, −15.58282894265242782220108482931, −14.43227975140991062688130503940, −13.37032717137252835153463281590, −12.56335847059551418628102232421, −11.5755966685438016410444198560, −11.111800355363761106597738610125, −10.24385580293910684032094169648, −9.59183923823910556105766531901, −8.988345499754905871845957019896, −7.71838583615548422651833494337, −6.88901475083150819274288592514, −6.11315455739025871631201138606, −4.580501624388676431173917033, −4.09326022831300169789093889274, −3.103512796359270123479010921304, −1.83860518989969603852043960862, −0.543133983185912269650421472023, 0.63834056944184916450429918479, 1.98166103695886832171710501297, 2.72051361333710145182614630301, 4.653431006499802230733036992606, 5.47531743464966047001112796891, 6.23940097924323607610354036619, 6.82616781147686839699290181997, 7.86224308827807148179608729226, 8.49737799494958707806993817785, 9.45951415363757770896813295798, 10.40614310250744346641779780032, 11.090180323418555611729403348994, 12.113564970339685491072758070796, 12.7226584275016859091642157146, 13.75052888253201312337886127152, 14.691042843712215731374227678193, 15.651155266161295051333383477270, 16.101828246068518336921774499357, 17.09286400847376851739920269393, 17.82519191614546828179734737597, 18.21558456745023617291365180207, 19.13507939550728595114854937967, 19.64575213348943241831601875981, 20.541683185899790604478734144282, 22.02549283848456409487192148511

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