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Properties

Label 1-149-149.123-r0-0-0
Degree $1$
Conductor $149$
Sign $0.674 - 0.738i$
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  + (0.660 − 0.750i)2-s + (0.0424 + 0.999i)3-s + (−0.127 − 0.991i)4-s + (0.942 − 0.333i)5-s + (0.778 + 0.628i)6-s + (0.372 − 0.927i)7-s + (−0.828 − 0.559i)8-s + (−0.996 + 0.0848i)9-s + (0.372 − 0.927i)10-s + (−0.127 + 0.991i)11-s + (0.985 − 0.169i)12-s + (0.524 − 0.851i)13-s + (−0.450 − 0.892i)14-s + (0.372 + 0.927i)15-s + (−0.967 + 0.251i)16-s + (0.0424 + 0.999i)17-s + ⋯
L(s)  = 1  + (0.660 − 0.750i)2-s + (0.0424 + 0.999i)3-s + (−0.127 − 0.991i)4-s + (0.942 − 0.333i)5-s + (0.778 + 0.628i)6-s + (0.372 − 0.927i)7-s + (−0.828 − 0.559i)8-s + (−0.996 + 0.0848i)9-s + (0.372 − 0.927i)10-s + (−0.127 + 0.991i)11-s + (0.985 − 0.169i)12-s + (0.524 − 0.851i)13-s + (−0.450 − 0.892i)14-s + (0.372 + 0.927i)15-s + (−0.967 + 0.251i)16-s + (0.0424 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.674 - 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.674 - 0.738i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.674 - 0.738i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.564440122 - 0.6898775606i\)
\(L(\frac12)\) \(\approx\) \(1.564440122 - 0.6898775606i\)
\(L(1)\) \(\approx\) \(1.502782800 - 0.4512739053i\)
\(L(1)\) \(\approx\) \(1.502782800 - 0.4512739053i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad149 \( 1 \)
good2 \( 1 + (0.660 - 0.750i)T \)
3 \( 1 + (0.0424 + 0.999i)T \)
5 \( 1 + (0.942 - 0.333i)T \)
7 \( 1 + (0.372 - 0.927i)T \)
11 \( 1 + (-0.127 + 0.991i)T \)
13 \( 1 + (0.524 - 0.851i)T \)
17 \( 1 + (0.0424 + 0.999i)T \)
19 \( 1 + (-0.594 - 0.803i)T \)
23 \( 1 + (0.524 + 0.851i)T \)
29 \( 1 + (0.210 - 0.977i)T \)
31 \( 1 + (0.524 + 0.851i)T \)
37 \( 1 + (-0.127 + 0.991i)T \)
41 \( 1 + (-0.967 + 0.251i)T \)
43 \( 1 + (-0.911 + 0.411i)T \)
47 \( 1 + (-0.594 + 0.803i)T \)
53 \( 1 + (-0.911 - 0.411i)T \)
59 \( 1 + (-0.292 + 0.956i)T \)
61 \( 1 + (0.660 - 0.750i)T \)
67 \( 1 + (-0.450 + 0.892i)T \)
71 \( 1 + (0.942 - 0.333i)T \)
73 \( 1 + (-0.721 + 0.691i)T \)
79 \( 1 + (-0.721 - 0.691i)T \)
83 \( 1 + (-0.721 + 0.691i)T \)
89 \( 1 + (0.873 - 0.487i)T \)
97 \( 1 + (-0.911 - 0.411i)T \)
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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

−28.566420498886524736651463480643, −26.92722025116359866038363064796, −25.861113034177333865875321841584, −25.04964698849317849431892467194, −24.52954068440255602799397887314, −23.514100591041813717064402965, −22.46575626492094927240575407904, −21.5051295570635873958689081308, −20.72314646837780590720637951249, −18.6470737058751409017634653677, −18.45480483072708430193639741059, −17.17957880636481609093727615777, −16.26174337880038905413952993272, −14.72699237631396226376145757549, −14.04297528928685580847397208209, −13.24098883708434857804550888385, −12.14314686726914977541260138200, −11.14867725409017810193889218519, −9.031337037244259794189552403551, −8.30481650542040359779697463754, −6.82781774776939691934293933394, −6.087612966294437749755186897259, −5.175119123311053838445947244739, −3.124885700087714917570367019472, −2.014197317724882381051038365400, 1.52893556428981031573083526456, 3.07111801341657507652543021246, 4.398824528498176495105027944102, 5.13481361345915880532297920847, 6.4100511925636393902262410924, 8.47214664942219531456386562825, 9.86041092490758161099642901172, 10.31900541728878310265653551487, 11.338243205763557270472507066946, 12.89177531715745079180350322367, 13.63443473056697088179703607855, 14.75876999264957864245225114868, 15.55099773175910079296390307674, 17.16364739743872404114459583248, 17.73631380642843565747641406992, 19.578096185083936390681397193702, 20.42245036391497155499787460786, 21.04594773593280486445079849955, 21.83226068987834385289851263940, 22.92922514780528629442830213918, 23.67040636884435369788540439998, 25.137972249212521755588982591991, 26.04571640297470225169159449140, 27.32894824853669912266944211921, 28.153429129495495099590528761676

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