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Properties

Label 1-1045-1045.137-r1-0-0
Degree $1$
Conductor $1045$
Sign $0.154 - 0.988i$
Analytic cond. $112.300$
Root an. cond. $112.300$
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  + (0.0697 − 0.997i)2-s + (−0.788 + 0.615i)3-s + (−0.990 − 0.139i)4-s + (0.559 + 0.829i)6-s + (−0.207 − 0.978i)7-s + (−0.207 + 0.978i)8-s + (0.241 − 0.970i)9-s + (0.866 − 0.5i)12-s + (−0.275 − 0.961i)13-s + (−0.990 + 0.139i)14-s + (0.961 + 0.275i)16-s + (0.970 − 0.241i)17-s + (−0.951 − 0.309i)18-s + (0.766 + 0.642i)21-s + (0.984 − 0.173i)23-s + (−0.438 − 0.898i)24-s + ⋯
L(s)  = 1  + (0.0697 − 0.997i)2-s + (−0.788 + 0.615i)3-s + (−0.990 − 0.139i)4-s + (0.559 + 0.829i)6-s + (−0.207 − 0.978i)7-s + (−0.207 + 0.978i)8-s + (0.241 − 0.970i)9-s + (0.866 − 0.5i)12-s + (−0.275 − 0.961i)13-s + (−0.990 + 0.139i)14-s + (0.961 + 0.275i)16-s + (0.970 − 0.241i)17-s + (−0.951 − 0.309i)18-s + (0.766 + 0.642i)21-s + (0.984 − 0.173i)23-s + (−0.438 − 0.898i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.154 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.154 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.154 - 0.988i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ 0.154 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.026561374 - 0.8788963808i\)
\(L(\frac12)\) \(\approx\) \(1.026561374 - 0.8788963808i\)
\(L(1)\) \(\approx\) \(0.7243794714 - 0.3692451195i\)
\(L(1)\) \(\approx\) \(0.7243794714 - 0.3692451195i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.0697 - 0.997i)T \)
3 \( 1 + (-0.788 + 0.615i)T \)
7 \( 1 + (-0.207 - 0.978i)T \)
13 \( 1 + (-0.275 - 0.961i)T \)
17 \( 1 + (0.970 - 0.241i)T \)
23 \( 1 + (0.984 - 0.173i)T \)
29 \( 1 + (0.374 + 0.927i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.951 + 0.309i)T \)
41 \( 1 + (-0.615 - 0.788i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (-0.529 + 0.848i)T \)
53 \( 1 + (-0.694 + 0.719i)T \)
59 \( 1 + (-0.848 + 0.529i)T \)
61 \( 1 + (0.438 - 0.898i)T \)
67 \( 1 + (0.642 + 0.766i)T \)
71 \( 1 + (-0.719 + 0.694i)T \)
73 \( 1 + (0.999 - 0.0348i)T \)
79 \( 1 + (-0.559 + 0.829i)T \)
83 \( 1 + (0.994 + 0.104i)T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (0.0697 - 0.997i)T \)
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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

−21.63847986127662327014426485736, −21.26473276025356259825973656367, −19.47588085267611320106147172745, −18.88612492718178066123935479532, −18.39691865925893718103713423244, −17.48597661213671568915954640033, −16.77854068403628325830326544095, −16.24411772817983211081326969251, −15.28074943248369889457704526584, −14.5895624896786951850153467057, −13.56813818043746913323103517275, −12.92539111316985773337216917110, −12.03949817670973273190577416539, −11.5158227140640008069033531927, −10.09792281262481365801015803377, −9.38149175207964931176032604840, −8.33074258868426917462471715552, −7.65607285926998538540016242124, −6.58060364338711567375202391060, −6.18947286815604854818971967903, −5.21275897885849684417040337369, −4.59217668522514985060007530208, −3.18856429856394494469805721791, −1.870786998005842339211314772973, −0.61249388361676138864047686084, 0.603906639072183152533593433672, 1.21526437278360457543460931349, 2.98755862267821214100856346702, 3.51349450734864965919042530785, 4.655669478208150711766225369118, 5.14186688230321396044816759816, 6.23538716149389172890471127517, 7.35338266762428857018331625809, 8.43828322700793992239356049326, 9.567664978002638216123410806, 10.15768462601846087619422917035, 10.70894049702861318023241392256, 11.47626786058210291656161750338, 12.40452443683382724616552263704, 12.952238365721484730942335139239, 13.99739864987530846051777421633, 14.762462827391751994965882889890, 15.73129512650901237846118938929, 16.8064514206316936608317526704, 17.224460961113079552743790458100, 18.062237210793613019911434445883, 18.876643401975543471853269555590, 19.858940703044373194512158841902, 20.49439189962247549348082620641, 21.09703078359262378604782458627

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