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Properties

Label 1-1441-1441.930-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.990 + 0.138i$
Analytic cond. $154.856$
Root an. cond. $154.856$
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  + (0.970 + 0.239i)2-s + (0.485 + 0.873i)3-s + (0.885 + 0.464i)4-s + (0.981 − 0.192i)5-s + (0.262 + 0.964i)6-s + (−0.836 − 0.548i)7-s + (0.748 + 0.663i)8-s + (−0.527 + 0.849i)9-s + (0.998 + 0.0483i)10-s + (0.0241 + 0.999i)12-s + (−0.779 + 0.626i)13-s + (−0.681 − 0.732i)14-s + (0.644 + 0.764i)15-s + (0.568 + 0.822i)16-s + (0.943 + 0.331i)17-s + (−0.715 + 0.698i)18-s + ⋯
L(s)  = 1  + (0.970 + 0.239i)2-s + (0.485 + 0.873i)3-s + (0.885 + 0.464i)4-s + (0.981 − 0.192i)5-s + (0.262 + 0.964i)6-s + (−0.836 − 0.548i)7-s + (0.748 + 0.663i)8-s + (−0.527 + 0.849i)9-s + (0.998 + 0.0483i)10-s + (0.0241 + 0.999i)12-s + (−0.779 + 0.626i)13-s + (−0.681 − 0.732i)14-s + (0.644 + 0.764i)15-s + (0.568 + 0.822i)16-s + (0.943 + 0.331i)17-s + (−0.715 + 0.698i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.990 + 0.138i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (930, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.990 + 0.138i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2799868730 + 4.027409175i\)
\(L(\frac12)\) \(\approx\) \(0.2799868730 + 4.027409175i\)
\(L(1)\) \(\approx\) \(1.789568705 + 1.221571980i\)
\(L(1)\) \(\approx\) \(1.789568705 + 1.221571980i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.970 + 0.239i)T \)
3 \( 1 + (0.485 + 0.873i)T \)
5 \( 1 + (0.981 - 0.192i)T \)
7 \( 1 + (-0.836 - 0.548i)T \)
13 \( 1 + (-0.779 + 0.626i)T \)
17 \( 1 + (0.943 + 0.331i)T \)
19 \( 1 + (-0.958 - 0.285i)T \)
23 \( 1 + (0.399 + 0.916i)T \)
29 \( 1 + (-0.644 + 0.764i)T \)
31 \( 1 + (-0.861 + 0.506i)T \)
37 \( 1 + (-0.989 + 0.144i)T \)
41 \( 1 + (0.607 + 0.794i)T \)
43 \( 1 + (-0.485 - 0.873i)T \)
47 \( 1 + (-0.0724 - 0.997i)T \)
53 \( 1 + T \)
59 \( 1 + (0.568 + 0.822i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.485 - 0.873i)T \)
71 \( 1 + (0.215 + 0.976i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.485 - 0.873i)T \)
83 \( 1 + (-0.715 - 0.698i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.998 + 0.0483i)T \)
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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

−20.39577704513901754526670451697, −19.212453309547706677896137376891, −19.061094080849763487223853770425, −18.11847168898588315245627534909, −17.11627824372200260739231821755, −16.41575013312521896580626841340, −15.19367238784127464776480654648, −14.6768449870774092886430589768, −14.06940675132656788614717379656, −13.11947217458540227618035428796, −12.72621183542810618454607486968, −12.21043738500039402242618622165, −11.09864418073277987853835737481, −10.057613522908158909551225628796, −9.54249377083505498337097173765, −8.44570214423750422811759773117, −7.299393862067398211797764625140, −6.70232262126731488290301950337, −5.80583995737342406483048306097, −5.44285424824743303955372453857, −3.97333873610328943365652596839, −2.89473470323332986666350527925, −2.52124675858359723397733873202, −1.648415071273728820794349999657, −0.40158518524663884740752840651, 1.60621977658297237382484857494, 2.50607054263607281034726424580, 3.41046722114346057886306049685, 4.05631975023332258072219249809, 5.12397569417563287799960900206, 5.580519823937834263411425504286, 6.71076243486882011129352414802, 7.33461808276275843925446033506, 8.57748467999209499355368059144, 9.37225769386682370692551562130, 10.19322748983638734442318358925, 10.70831620515959630324169286182, 11.84612159666737065734698404239, 12.88349465128824482222062812769, 13.31406491565893078440924993135, 14.22257855815184348375136284221, 14.6230614326239025186440131331, 15.47706841272770638379449686511, 16.488659556017477047487246535168, 16.74809336212405434208854402832, 17.420529912253901542131201768927, 18.95474742988063018208336889746, 19.705983900197789561815469497554, 20.31619408464237932146289978265, 21.18533561943177533040046534643

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