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Properties

Label 1-149-149.19-r0-0-0
Degree $1$
Conductor $149$
Sign $0.988 - 0.153i$
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  + (−0.911 − 0.411i)2-s + (−0.721 + 0.691i)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 + (−0.292 − 0.956i)8-s + (0.0424 − 0.999i)9-s + (−0.828 − 0.559i)10-s + (0.660 − 0.750i)11-s + (−0.996 − 0.0848i)12-s + (0.873 + 0.487i)13-s + (0.524 + 0.851i)14-s + (−0.828 + 0.559i)15-s + (−0.127 + 0.991i)16-s + (−0.721 + 0.691i)17-s + ⋯
L(s)  = 1  + (−0.911 − 0.411i)2-s + (−0.721 + 0.691i)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 + (−0.292 − 0.956i)8-s + (0.0424 − 0.999i)9-s + (−0.828 − 0.559i)10-s + (0.660 − 0.750i)11-s + (−0.996 − 0.0848i)12-s + (0.873 + 0.487i)13-s + (0.524 + 0.851i)14-s + (−0.828 + 0.559i)15-s + (−0.127 + 0.991i)16-s + (−0.721 + 0.691i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 149 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(149\)
Sign: $0.988 - 0.153i$
Analytic conductor: \(0.691953\)
Root analytic conductor: \(0.691953\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{149} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 149,\ (0:\ ),\ 0.988 - 0.153i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6596236274 - 0.05104316526i\)
\(L(\frac12)\) \(\approx\) \(0.6596236274 - 0.05104316526i\)
\(L(1)\) \(\approx\) \(0.6723185347 + 0.02229127299i\)
\(L(1)\) \(\approx\) \(0.6723185347 + 0.02229127299i\)

Euler product

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

−28.26563457053812538156026748700, −27.39593614854679670765884592047, −25.86386196759616441339722770415, −25.04257018939208621552556936477, −24.8254830179951561866380411730, −23.21764873396147660260064370891, −22.56504456907918500090149254349, −21.15062991281334875470632000504, −19.88546539834073274809909185918, −18.85499500099351011217120187429, −18.01550080183469897976996153526, −17.31523735677622451869099936697, −16.39489225501462509809894920471, −15.34622949302398598040425314760, −13.84346302488856344828623475337, −12.77913824799280712495301176860, −11.647122752719819728295671637121, −10.36928342175676124965425073934, −9.46303305679018766101289973597, −8.33516528986400675667658764588, −6.74138719058820033257836076332, −6.27259306519465648202039915898, −5.16483703091647881995859415795, −2.467582354792341090700368053457, −1.23238248175638403000179503343, 1.06348052155484850259180911021, 2.958641451398025078772820128628, 4.228879722731413054213784545, 6.288477463430615489809315681663, 6.60181230876697835017339398270, 8.82398627868583495752882185397, 9.447303253330372780909189247631, 10.66403304573250146680463853616, 11.07353897757236200105681340850, 12.60590311779297834727063851425, 13.678283731409853623759971808103, 15.35567337384724377285629083755, 16.50893650608963945244510664408, 17.014697899095866698902558824999, 17.943120993000365529447235590129, 19.07113096513418582612204778465, 20.15714470499892778084963703707, 21.365962495629275022939449796108, 21.79952722938198142784110920696, 22.91772828077885020136414055848, 24.313937243711417872552544362546, 25.63256785400837136017406347447, 26.31384626603034810474372271424, 27.056394683954126296570790558601, 28.357880422495315317969078920350

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