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Properties

Label 1-149-149.118-r0-0-0
Degree $1$
Conductor $149$
Sign $0.930 - 0.366i$
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  + (−0.778 + 0.628i)2-s + (0.828 + 0.559i)3-s + (0.210 − 0.977i)4-s + (0.0424 − 0.999i)5-s + (−0.996 + 0.0848i)6-s + (−0.594 − 0.803i)7-s + (0.450 + 0.892i)8-s + (0.372 + 0.927i)9-s + (0.594 + 0.803i)10-s + (−0.210 − 0.977i)11-s + (0.721 − 0.691i)12-s + (0.127 − 0.991i)13-s + (0.967 + 0.251i)14-s + (0.594 − 0.803i)15-s + (−0.911 − 0.411i)16-s + (−0.828 − 0.559i)17-s + ⋯
L(s)  = 1  + (−0.778 + 0.628i)2-s + (0.828 + 0.559i)3-s + (0.210 − 0.977i)4-s + (0.0424 − 0.999i)5-s + (−0.996 + 0.0848i)6-s + (−0.594 − 0.803i)7-s + (0.450 + 0.892i)8-s + (0.372 + 0.927i)9-s + (0.594 + 0.803i)10-s + (−0.210 − 0.977i)11-s + (0.721 − 0.691i)12-s + (0.127 − 0.991i)13-s + (0.967 + 0.251i)14-s + (0.594 − 0.803i)15-s + (−0.911 − 0.411i)16-s + (−0.828 − 0.559i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.930 - 0.366i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.930 - 0.366i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8823950274 - 0.1675638924i\)
\(L(\frac12)\) \(\approx\) \(0.8823950274 - 0.1675638924i\)
\(L(1)\) \(\approx\) \(0.8910533816 + 0.02591775098i\)
\(L(1)\) \(\approx\) \(0.8910533816 + 0.02591775098i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (-0.778 + 0.628i)T \)
3 \( 1 + (0.828 + 0.559i)T \)
5 \( 1 + (0.0424 - 0.999i)T \)
7 \( 1 + (-0.594 - 0.803i)T \)
11 \( 1 + (-0.210 - 0.977i)T \)
13 \( 1 + (0.127 - 0.991i)T \)
17 \( 1 + (-0.828 - 0.559i)T \)
19 \( 1 + (0.873 - 0.487i)T \)
23 \( 1 + (0.127 + 0.991i)T \)
29 \( 1 + (0.985 + 0.169i)T \)
31 \( 1 + (-0.127 - 0.991i)T \)
37 \( 1 + (0.210 + 0.977i)T \)
41 \( 1 + (0.911 + 0.411i)T \)
43 \( 1 + (-0.942 + 0.333i)T \)
47 \( 1 + (0.873 + 0.487i)T \)
53 \( 1 + (0.942 + 0.333i)T \)
59 \( 1 + (-0.524 + 0.851i)T \)
61 \( 1 + (0.778 - 0.628i)T \)
67 \( 1 + (-0.967 + 0.251i)T \)
71 \( 1 + (-0.0424 + 0.999i)T \)
73 \( 1 + (-0.292 - 0.956i)T \)
79 \( 1 + (0.292 - 0.956i)T \)
83 \( 1 + (0.292 + 0.956i)T \)
89 \( 1 + (-0.660 - 0.750i)T \)
97 \( 1 + (-0.942 - 0.333i)T \)
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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

−28.52629838679013906233041035126, −26.859312936902018131054609362584, −26.30637553232775182885631826524, −25.50333020466566918771209138322, −24.7429318619917716471574139931, −23.12052643320190457450531653656, −22.01724573222226023478134054658, −21.11398197157857154727443193977, −19.91688882961719866835386301586, −19.18565245007874583735398374656, −18.35799088428370097812777573188, −17.7852198484741070181383907422, −16.065834792915967802112580812288, −15.05060345917827224292742183695, −13.87148281674026242988428908957, −12.63316516308891289726468908410, −11.87229074197337564202005977225, −10.422695803762735356815244711551, −9.449224882134498096994061962913, −8.53595275403090753639589276047, −7.21390153025321131759528414849, −6.51258724030121211704052936269, −3.92620817935459990895927367342, −2.6924810030819741886988170083, −1.956516769678091394455622157531, 0.953828626324515929193239908307, 2.96767129059160516685126485601, 4.55801261360009842554238425629, 5.73829157332435419647500909303, 7.38501483778595541204477044831, 8.3013789560654537068533359472, 9.24822726949506156489323511367, 10.052985948797286019000583495106, 11.21918002278048085602785127939, 13.34677995846355966551833190178, 13.73259358243373493049413987518, 15.36649891870551195722354310015, 16.01711028495532697230912913503, 16.76650632249048972577917418162, 17.93469327621067096467816042219, 19.36733049371702540655945912076, 20.014545776717344662496247998342, 20.681496323281016254565458694318, 22.12628718778853658231950832473, 23.487250290020454204903945976563, 24.49752146885091858033012967470, 25.19616833801581357664155320224, 26.20842640221428940341666484510, 26.976609250828610974762544753874, 27.700761975544755588120701014220

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