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Properties

Label 1-1049-1049.6-r1-0-0
Degree $1$
Conductor $1049$
Sign $-0.0596 + 0.998i$
Analytic cond. $112.730$
Root an. cond. $112.730$
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  + (−0.995 − 0.0957i)2-s + (−0.905 − 0.423i)3-s + (0.981 + 0.190i)4-s + (0.461 − 0.887i)5-s + (0.860 + 0.508i)6-s + (−0.0778 + 0.996i)7-s + (−0.958 − 0.283i)8-s + (0.640 + 0.767i)9-s + (−0.544 + 0.838i)10-s + (0.744 + 0.667i)11-s + (−0.808 − 0.588i)12-s + (0.0119 − 0.999i)13-s + (0.172 − 0.984i)14-s + (−0.793 + 0.607i)15-s + (0.927 + 0.374i)16-s + (0.125 + 0.992i)17-s + ⋯
L(s)  = 1  + (−0.995 − 0.0957i)2-s + (−0.905 − 0.423i)3-s + (0.981 + 0.190i)4-s + (0.461 − 0.887i)5-s + (0.860 + 0.508i)6-s + (−0.0778 + 0.996i)7-s + (−0.958 − 0.283i)8-s + (0.640 + 0.767i)9-s + (−0.544 + 0.838i)10-s + (0.744 + 0.667i)11-s + (−0.808 − 0.588i)12-s + (0.0119 − 0.999i)13-s + (0.172 − 0.984i)14-s + (−0.793 + 0.607i)15-s + (0.927 + 0.374i)16-s + (0.125 + 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0596 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1049 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0596 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1049\)
Sign: $-0.0596 + 0.998i$
Analytic conductor: \(112.730\)
Root analytic conductor: \(112.730\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1049} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1049,\ (1:\ ),\ -0.0596 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4314833195 + 0.4580190308i\)
\(L(\frac12)\) \(\approx\) \(0.4314833195 + 0.4580190308i\)
\(L(1)\) \(\approx\) \(0.5731486184 + 0.02801675609i\)
\(L(1)\) \(\approx\) \(0.5731486184 + 0.02801675609i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1049 \( 1 \)
good2 \( 1 + (-0.995 - 0.0957i)T \)
3 \( 1 + (-0.905 - 0.423i)T \)
5 \( 1 + (0.461 - 0.887i)T \)
7 \( 1 + (-0.0778 + 0.996i)T \)
11 \( 1 + (0.744 + 0.667i)T \)
13 \( 1 + (0.0119 - 0.999i)T \)
17 \( 1 + (0.125 + 0.992i)T \)
19 \( 1 + (-0.851 - 0.524i)T \)
23 \( 1 + (-0.379 + 0.925i)T \)
29 \( 1 + (0.554 + 0.832i)T \)
31 \( 1 + (0.872 + 0.487i)T \)
37 \( 1 + (0.966 - 0.254i)T \)
41 \( 1 + (0.972 - 0.231i)T \)
43 \( 1 + (-0.752 - 0.658i)T \)
47 \( 1 + (-0.0898 + 0.995i)T \)
53 \( 1 + (-0.0599 + 0.998i)T \)
59 \( 1 + (0.818 + 0.574i)T \)
61 \( 1 + (0.685 - 0.727i)T \)
67 \( 1 + (-0.00599 + 0.999i)T \)
71 \( 1 + (-0.508 - 0.860i)T \)
73 \( 1 + (-0.908 + 0.418i)T \)
79 \( 1 + (-0.727 + 0.685i)T \)
83 \( 1 + (0.539 + 0.842i)T \)
89 \( 1 + (0.996 - 0.0778i)T \)
97 \( 1 + (-0.971 - 0.237i)T \)
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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

−21.16572380375928818526234426743, −20.39261252509434677917342818760, −19.22529243464382786486969911372, −18.78476597320139552817524301579, −17.85731831728936446351437273616, −17.26646654679924843309783473247, −16.46099193285598054724632872629, −16.21687289160717953238546560365, −14.85387524664969721098289819733, −14.33201833862385370196021633993, −13.28353493067961206711095724447, −11.77149777791761727808320072953, −11.47739697629455882692094939659, −10.58899334601855939700674237088, −9.97896692856587684429960447319, −9.401111549549036742704668606374, −8.19702551653370366263071358427, −7.068746916625810968563640146225, −6.46485444006660030707287125645, −6.05224776300734950103743472440, −4.54580438253960696282683782262, −3.62118416300417928093249059138, −2.39337217023291555289227471936, −1.16951574498686468057016865049, −0.251649139784894535669436516380, 1.01311819992959874402555486442, 1.693931456081932627272426697799, 2.64684297854836901433071802541, 4.27583588708890884861607551648, 5.46111917987049912117235469434, 6.02300467584493990002802596996, 6.82958212817159269287854863121, 7.97323075126759416178925068846, 8.631168825469948419767063889910, 9.52689689835135054018984932615, 10.25392060332742831930509487597, 11.14667641703737566211508207957, 12.16923088534633582574048134076, 12.44686626698755210672231189181, 13.2058816914897615443574576084, 14.75798460183368092732171468567, 15.60986643132843176913159962996, 16.24249690558166745604881205455, 17.28303455612051004680538238519, 17.52738553984004911984956656770, 18.12796564322970754706070491049, 19.282929904968039289967946144526, 19.66200192317112927229290119258, 20.6784814665062658687600197324, 21.71542638063304301246549880893

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