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Properties

Label 1-4011-4011.59-r0-0-0
Degree $1$
Conductor $4011$
Sign $-0.897 - 0.441i$
Analytic cond. $18.6270$
Root an. cond. $18.6270$
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  + (0.0605 − 0.998i)2-s + (−0.992 − 0.120i)4-s + (−0.754 + 0.656i)5-s + (−0.180 + 0.983i)8-s + (0.609 + 0.792i)10-s + (0.962 − 0.272i)11-s + (0.627 − 0.778i)13-s + (0.970 + 0.240i)16-s + (−0.126 − 0.991i)17-s + (0.609 − 0.792i)19-s + (0.828 − 0.560i)20-s + (−0.213 − 0.976i)22-s + (−0.565 − 0.824i)23-s + (0.137 − 0.990i)25-s + (−0.739 − 0.673i)26-s + ⋯
L(s)  = 1  + (0.0605 − 0.998i)2-s + (−0.992 − 0.120i)4-s + (−0.754 + 0.656i)5-s + (−0.180 + 0.983i)8-s + (0.609 + 0.792i)10-s + (0.962 − 0.272i)11-s + (0.627 − 0.778i)13-s + (0.970 + 0.240i)16-s + (−0.126 − 0.991i)17-s + (0.609 − 0.792i)19-s + (0.828 − 0.560i)20-s + (−0.213 − 0.976i)22-s + (−0.565 − 0.824i)23-s + (0.137 − 0.990i)25-s + (−0.739 − 0.673i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $-0.897 - 0.441i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ -0.897 - 0.441i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2771541916 - 1.189623716i\)
\(L(\frac12)\) \(\approx\) \(0.2771541916 - 1.189623716i\)
\(L(1)\) \(\approx\) \(0.7645448446 - 0.5164787391i\)
\(L(1)\) \(\approx\) \(0.7645448446 - 0.5164787391i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (0.0605 - 0.998i)T \)
5 \( 1 + (-0.754 + 0.656i)T \)
11 \( 1 + (0.962 - 0.272i)T \)
13 \( 1 + (0.627 - 0.778i)T \)
17 \( 1 + (-0.126 - 0.991i)T \)
19 \( 1 + (0.609 - 0.792i)T \)
23 \( 1 + (-0.565 - 0.824i)T \)
29 \( 1 + (0.340 + 0.940i)T \)
31 \( 1 + (0.592 - 0.805i)T \)
37 \( 1 + (-0.592 - 0.805i)T \)
41 \( 1 + (-0.986 + 0.164i)T \)
43 \( 1 + (0.490 + 0.871i)T \)
47 \( 1 + (0.884 - 0.466i)T \)
53 \( 1 + (-0.618 + 0.785i)T \)
59 \( 1 + (0.00551 - 0.999i)T \)
61 \( 1 + (0.480 + 0.876i)T \)
67 \( 1 + (0.287 + 0.957i)T \)
71 \( 1 + (0.148 - 0.988i)T \)
73 \( 1 + (0.441 + 0.897i)T \)
79 \( 1 + (-0.644 - 0.764i)T \)
83 \( 1 + (0.431 + 0.901i)T \)
89 \( 1 + (0.0935 - 0.995i)T \)
97 \( 1 + (0.995 + 0.0990i)T \)
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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

−18.96556187469785238384636021664, −17.83406603511048140155242836465, −17.18665107729757753513256769643, −16.76134669876119871860003245923, −15.926643305256307081393937918685, −15.55564931344987056715829501925, −14.83384801282455292291842900417, −13.92832254581434599553293351380, −13.64642957623048567622023235596, −12.53545445376650408763927933423, −12.08416196009319265692385029006, −11.449272070719235311636346693375, −10.27404514384588471421935606211, −9.54339831040967654993735722423, −8.80959946096582779264314312968, −8.3029368273722427974208688671, −7.64391293150462221830850080678, −6.786406556156148557962380928427, −6.2069299649311935648487208517, −5.37943918137651223521300497386, −4.547453738154724326212844332976, −3.85444025334374024681931643655, −3.51022945410677948382294939300, −1.7267447201444451119304702109, −1.050378319445445558881418778873, 0.4348578373564437446158423705, 1.1815387942275184516043873588, 2.44174676840519262508671287604, 3.03393619761930401207699363405, 3.71609791142657112623367712806, 4.39704717481123923565511338437, 5.23016230588942579096178874543, 6.18379337731371967876999561831, 6.97887257043175640546659035141, 7.841696673251602027788751564233, 8.597142452967342687900876520983, 9.20418312408541430103573932799, 10.078823335102123413992742576245, 10.74759922333268825943274833364, 11.363015652940491110931327545470, 11.84566665780232329431406479208, 12.51744024958683622555328864828, 13.39116921216145981818695690911, 14.09221348008929495655098661493, 14.52087604231066079416372220454, 15.474172142641761510820602554017, 16.02386579453238641929205923751, 16.97657459301744449534417758044, 17.88038193493244903900470818807, 18.24727569819102026821521127309

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