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Properties

Label 1-4011-4011.32-r1-0-0
Degree $1$
Conductor $4011$
Sign $0.580 - 0.813i$
Analytic cond. $431.041$
Root an. cond. $431.041$
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  + (0.191 − 0.981i)2-s + (−0.926 − 0.376i)4-s + (−0.137 − 0.990i)5-s + (−0.546 + 0.837i)8-s + (−0.998 − 0.0550i)10-s + (−0.851 − 0.523i)11-s + (−0.401 − 0.915i)13-s + (0.716 + 0.697i)16-s + (−0.635 − 0.771i)17-s + (−0.998 + 0.0550i)19-s + (−0.245 + 0.969i)20-s + (−0.677 + 0.735i)22-s + (0.998 − 0.0550i)23-s + (−0.962 + 0.272i)25-s + (−0.975 + 0.218i)26-s + ⋯
L(s)  = 1  + (0.191 − 0.981i)2-s + (−0.926 − 0.376i)4-s + (−0.137 − 0.990i)5-s + (−0.546 + 0.837i)8-s + (−0.998 − 0.0550i)10-s + (−0.851 − 0.523i)11-s + (−0.401 − 0.915i)13-s + (0.716 + 0.697i)16-s + (−0.635 − 0.771i)17-s + (−0.998 + 0.0550i)19-s + (−0.245 + 0.969i)20-s + (−0.677 + 0.735i)22-s + (0.998 − 0.0550i)23-s + (−0.962 + 0.272i)25-s + (−0.975 + 0.218i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.580 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.580 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $0.580 - 0.813i$
Analytic conductor: \(431.041\)
Root analytic conductor: \(431.041\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (1:\ ),\ 0.580 - 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1330872555 - 0.06852168475i\)
\(L(\frac12)\) \(\approx\) \(0.1330872555 - 0.06852168475i\)
\(L(1)\) \(\approx\) \(0.4666376882 - 0.5521805172i\)
\(L(1)\) \(\approx\) \(0.4666376882 - 0.5521805172i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.191 - 0.981i)T \)
5 \( 1 + (-0.137 - 0.990i)T \)
11 \( 1 + (-0.851 - 0.523i)T \)
13 \( 1 + (-0.401 - 0.915i)T \)
17 \( 1 + (-0.635 - 0.771i)T \)
19 \( 1 + (-0.998 + 0.0550i)T \)
23 \( 1 + (0.998 - 0.0550i)T \)
29 \( 1 + (-0.245 - 0.969i)T \)
31 \( 1 + (-0.298 + 0.954i)T \)
37 \( 1 + (-0.298 - 0.954i)T \)
41 \( 1 + (-0.945 - 0.324i)T \)
43 \( 1 + (-0.0825 + 0.996i)T \)
47 \( 1 + (-0.0275 + 0.999i)T \)
53 \( 1 + (-0.851 - 0.523i)T \)
59 \( 1 + (0.298 - 0.954i)T \)
61 \( 1 + (0.635 - 0.771i)T \)
67 \( 1 + (0.350 + 0.936i)T \)
71 \( 1 + (-0.945 - 0.324i)T \)
73 \( 1 + (0.0275 + 0.999i)T \)
79 \( 1 + (0.716 + 0.697i)T \)
83 \( 1 + (-0.546 + 0.837i)T \)
89 \( 1 + (-0.904 - 0.426i)T \)
97 \( 1 + (-0.677 + 0.735i)T \)
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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

−18.43076198676198286054640720411, −17.59768422114369628570947401583, −17.03284665142613460727287125069, −16.40626239308616134904055594306, −15.38364584462556975005984264570, −15.06200969558199724209299932493, −14.65103108743552892242693848287, −13.6450429413165627427707493341, −13.20457340060377031231641149491, −12.40865945224259812698891125938, −11.58898815898155575205750108738, −10.66220907610239829958275665118, −10.16466696064589896273763941465, −9.21978801793053015697087915612, −8.55231568855184170964209765322, −7.762976810917473154043709379899, −6.958033743017316599595178143073, −6.72494166077646846045815235324, −5.81641104416304655941630224785, −4.94170318235955856825540506178, −4.27849940671088021312576033891, −3.49761182089665940496201598202, −2.588007432493138190882493004011, −1.73551513019216789580064792336, −0.04519811251439323309874108272, 0.39140420987147420755983408032, 1.321064078009860343909524585458, 2.31592094817912649381071496082, 2.95843539736996836929399130958, 3.85426491831107102726226714425, 4.7293276357335442166659342441, 5.186525523324416082331857345001, 5.843187712612187065434340465, 7.01917280167318235078179000326, 8.184119222147786988507446198458, 8.4005950378911550191618797285, 9.350619647997699408656294848857, 9.87565962302689522840598019533, 10.88286951060094607357788581706, 11.1624869385504234176778526296, 12.16211691870791785182449942022, 12.87245737859536424169713959596, 13.044040736070876820417013041158, 13.872222972433911326627659014970, 14.70382486572704306035949728008, 15.528866912976253847409710826083, 16.069913939560620201080618425718, 17.10968559579072289945339763090, 17.545091614128419388046741291856, 18.28215525173456304791945993252

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