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Properties

Label 1-149-149.18-r1-0-0
Degree $1$
Conductor $149$
Sign $0.991 - 0.131i$
Analytic cond. $16.0122$
Root an. cond. $16.0122$
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  + (0.411 + 0.911i)2-s + (0.691 − 0.721i)3-s + (−0.660 + 0.750i)4-s + (0.985 − 0.169i)5-s + (0.942 + 0.333i)6-s + (0.828 − 0.559i)7-s + (−0.956 − 0.292i)8-s + (−0.0424 − 0.999i)9-s + (0.559 + 0.828i)10-s + (0.750 − 0.660i)11-s + (0.0848 + 0.996i)12-s + (−0.487 − 0.873i)13-s + (0.851 + 0.524i)14-s + (0.559 − 0.828i)15-s + (−0.127 − 0.991i)16-s + (−0.721 − 0.691i)17-s + ⋯
L(s)  = 1  + (0.411 + 0.911i)2-s + (0.691 − 0.721i)3-s + (−0.660 + 0.750i)4-s + (0.985 − 0.169i)5-s + (0.942 + 0.333i)6-s + (0.828 − 0.559i)7-s + (−0.956 − 0.292i)8-s + (−0.0424 − 0.999i)9-s + (0.559 + 0.828i)10-s + (0.750 − 0.660i)11-s + (0.0848 + 0.996i)12-s + (−0.487 − 0.873i)13-s + (0.851 + 0.524i)14-s + (0.559 − 0.828i)15-s + (−0.127 − 0.991i)16-s + (−0.721 − 0.691i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.991 - 0.131i$
Analytic conductor: \(16.0122\)
Root analytic conductor: \(16.0122\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (1:\ ),\ 0.991 - 0.131i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.241076913 - 0.2134341374i\)
\(L(\frac12)\) \(\approx\) \(3.241076913 - 0.2134341374i\)
\(L(1)\) \(\approx\) \(1.897584181 + 0.1442944137i\)
\(L(1)\) \(\approx\) \(1.897584181 + 0.1442944137i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (0.411 + 0.911i)T \)
3 \( 1 + (0.691 - 0.721i)T \)
5 \( 1 + (0.985 - 0.169i)T \)
7 \( 1 + (0.828 - 0.559i)T \)
11 \( 1 + (0.750 - 0.660i)T \)
13 \( 1 + (-0.487 - 0.873i)T \)
17 \( 1 + (-0.721 - 0.691i)T \)
19 \( 1 + (-0.450 + 0.892i)T \)
23 \( 1 + (-0.487 + 0.873i)T \)
29 \( 1 + (0.778 - 0.628i)T \)
31 \( 1 + (0.873 + 0.487i)T \)
37 \( 1 + (0.660 + 0.750i)T \)
41 \( 1 + (-0.991 + 0.127i)T \)
43 \( 1 + (0.977 - 0.210i)T \)
47 \( 1 + (0.450 + 0.892i)T \)
53 \( 1 + (-0.210 + 0.977i)T \)
59 \( 1 + (-0.803 + 0.594i)T \)
61 \( 1 + (0.911 - 0.411i)T \)
67 \( 1 + (0.524 + 0.851i)T \)
71 \( 1 + (0.169 + 0.985i)T \)
73 \( 1 + (0.372 + 0.927i)T \)
79 \( 1 + (-0.927 - 0.372i)T \)
83 \( 1 + (-0.927 + 0.372i)T \)
89 \( 1 + (-0.251 - 0.967i)T \)
97 \( 1 + (-0.977 - 0.210i)T \)
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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

−28.14460369872750837454406773032, −27.02944570521373941107002489461, −26.10476049481296971251548095604, −24.9223467257547945893199696446, −24.053891442346673340296662752306, −22.35188994085828439945860159266, −21.77633138096955064180746390396, −21.153365283423318828922409300327, −20.159156616988135940795925096860, −19.25361483663120962250694672814, −18.0288557659344541499241859690, −17.08644309964407528694983657427, −15.26087650084871834845119105178, −14.51614048811732296235419447014, −13.85826131280819549846506070534, −12.587231553945222888787382771979, −11.314934612854243660180361144202, −10.30861579574793651918844965285, −9.31415072496485286775665214197, −8.62078005218491523175867400405, −6.46997358510431642861364442911, −4.95603895199231190003312451643, −4.237040267538209610967063325761, −2.46944398980009773942627633679, −1.87855252698456166411536306707, 1.0932602404416872629738397441, 2.756728633078102364868226867641, 4.279784939314456240104511526013, 5.70169562026295650036712977053, 6.70209421759736266214129941396, 7.87180397925785110415024505249, 8.70080925522066532707289393712, 9.905919070847746551674640459416, 11.7964517398290195439807692117, 12.98284732107299836968139499633, 13.948211539267328303131889568535, 14.26584155113298341557982362689, 15.54022112657527534028107527433, 17.15000665449222269092247855205, 17.53500579403709800940382473617, 18.59551799886459780093340103702, 20.0625129538225466059674981910, 21.01388872138731725681477284130, 21.99172358126167650604829806210, 23.21849373147006510351124133480, 24.29819568158654394209658811900, 24.857144247038850658741618726653, 25.49032191588307683886582456633, 26.725811893444034747054774557, 27.36809335434442175320423244660

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