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Properties

Label 1-149-149.28-r0-0-0
Degree $1$
Conductor $149$
Sign $0.947 - 0.319i$
Analytic cond. $0.691953$
Root an. cond. $0.691953$
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  + (0.985 − 0.169i)2-s + (−0.594 − 0.803i)3-s + (0.942 − 0.333i)4-s + (0.372 + 0.927i)5-s + (−0.721 − 0.691i)6-s + (0.524 + 0.851i)7-s + (0.873 − 0.487i)8-s + (−0.292 + 0.956i)9-s + (0.524 + 0.851i)10-s + (0.942 + 0.333i)11-s + (−0.828 − 0.559i)12-s + (−0.911 − 0.411i)13-s + (0.660 + 0.750i)14-s + (0.524 − 0.851i)15-s + (0.778 − 0.628i)16-s + (−0.594 − 0.803i)17-s + ⋯
L(s)  = 1  + (0.985 − 0.169i)2-s + (−0.594 − 0.803i)3-s + (0.942 − 0.333i)4-s + (0.372 + 0.927i)5-s + (−0.721 − 0.691i)6-s + (0.524 + 0.851i)7-s + (0.873 − 0.487i)8-s + (−0.292 + 0.956i)9-s + (0.524 + 0.851i)10-s + (0.942 + 0.333i)11-s + (−0.828 − 0.559i)12-s + (−0.911 − 0.411i)13-s + (0.660 + 0.750i)14-s + (0.524 − 0.851i)15-s + (0.778 − 0.628i)16-s + (−0.594 − 0.803i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.947 - 0.319i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (28, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.947 - 0.319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.785496967 - 0.2927066118i\)
\(L(\frac12)\) \(\approx\) \(1.785496967 - 0.2927066118i\)
\(L(1)\) \(\approx\) \(1.636622640 - 0.2359832491i\)
\(L(1)\) \(\approx\) \(1.636622640 - 0.2359832491i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (0.985 - 0.169i)T \)
3 \( 1 + (-0.594 - 0.803i)T \)
5 \( 1 + (0.372 + 0.927i)T \)
7 \( 1 + (0.524 + 0.851i)T \)
11 \( 1 + (0.942 + 0.333i)T \)
13 \( 1 + (-0.911 - 0.411i)T \)
17 \( 1 + (-0.594 - 0.803i)T \)
19 \( 1 + (-0.127 - 0.991i)T \)
23 \( 1 + (-0.911 + 0.411i)T \)
29 \( 1 + (0.0424 - 0.999i)T \)
31 \( 1 + (-0.911 + 0.411i)T \)
37 \( 1 + (0.942 + 0.333i)T \)
41 \( 1 + (0.778 - 0.628i)T \)
43 \( 1 + (-0.996 + 0.0848i)T \)
47 \( 1 + (-0.127 + 0.991i)T \)
53 \( 1 + (-0.996 - 0.0848i)T \)
59 \( 1 + (-0.967 + 0.251i)T \)
61 \( 1 + (0.985 - 0.169i)T \)
67 \( 1 + (0.660 - 0.750i)T \)
71 \( 1 + (0.372 + 0.927i)T \)
73 \( 1 + (-0.450 - 0.892i)T \)
79 \( 1 + (-0.450 + 0.892i)T \)
83 \( 1 + (-0.450 - 0.892i)T \)
89 \( 1 + (0.210 - 0.977i)T \)
97 \( 1 + (-0.996 - 0.0848i)T \)
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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

−28.28912507494560427705074001132, −27.21844543680936742334860860497, −26.24734573719914567928722608408, −24.887442526176905374711302628689, −24.05398089111663777496830992738, −23.34602424885451908209960899427, −21.995345956106743563422538169175, −21.61237949653914110252842585552, −20.41066953120134197692113953550, −19.89512728672994703968304251494, −17.61110621123355001974727222350, −16.6846638832489905085967793225, −16.425040642052726811378795888, −14.801769558969109864346712134690, −14.18078168697210077491585006047, −12.79595076543166847386498287056, −11.88037912061439753879947787893, −10.86527179874972397133507839629, −9.70253987850177776783546634268, −8.24937837147133519742209496002, −6.651179061630354411130672833133, −5.59896042777057989453178411075, −4.473546408370237551590589908509, −3.88578234661423159403855227385, −1.65788271593377945303557312607, 1.90017721275896184872855814369, 2.72832607277281570437467557596, 4.67209959853347825022650373317, 5.78189472214969683374504352774, 6.69586849056647831669271336582, 7.62199119534858531397058087751, 9.64569193188843455116684515282, 11.13145856944837440188773025836, 11.67891667141755438019804265214, 12.68766032144778286742457393394, 13.8634349333233666352189703500, 14.659860225630813791931452191468, 15.6741090693220987890824043885, 17.26558340026811224402135322454, 18.04185847849004486660627131178, 19.18955268060533549041020476624, 20.081318681856927226494092401092, 21.79351003435687440031800104304, 22.09668391002445589277767084421, 22.96277762545033438897121544334, 24.17771835693889235264162215953, 24.85393249886188936336993440047, 25.61399688015494331117813005241, 27.33014124647476288523926379285, 28.41129894528058290182289104772

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