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Properties

Label 1-66e2-4356.443-r0-0-0
Degree $1$
Conductor $4356$
Sign $0.00403 + 0.999i$
Analytic cond. $20.2291$
Root an. cond. $20.2291$
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  + (−0.345 + 0.938i)5-s + (0.683 + 0.730i)7-s + (0.953 − 0.299i)13-s + (0.0285 + 0.999i)17-s + (−0.610 + 0.791i)19-s + (0.981 − 0.189i)23-s + (−0.761 − 0.647i)25-s + (0.179 + 0.983i)29-s + (0.398 − 0.917i)31-s + (−0.921 + 0.389i)35-s + (0.198 − 0.980i)37-s + (0.625 + 0.780i)41-s + (0.786 − 0.618i)43-s + (0.548 + 0.836i)47-s + (−0.0665 + 0.997i)49-s + ⋯
L(s)  = 1  + (−0.345 + 0.938i)5-s + (0.683 + 0.730i)7-s + (0.953 − 0.299i)13-s + (0.0285 + 0.999i)17-s + (−0.610 + 0.791i)19-s + (0.981 − 0.189i)23-s + (−0.761 − 0.647i)25-s + (0.179 + 0.983i)29-s + (0.398 − 0.917i)31-s + (−0.921 + 0.389i)35-s + (0.198 − 0.980i)37-s + (0.625 + 0.780i)41-s + (0.786 − 0.618i)43-s + (0.548 + 0.836i)47-s + (−0.0665 + 0.997i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00403 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00403 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4356\)    =    \(2^{2} \cdot 3^{2} \cdot 11^{2}\)
Sign: $0.00403 + 0.999i$
Analytic conductor: \(20.2291\)
Root analytic conductor: \(20.2291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4356} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4356,\ (0:\ ),\ 0.00403 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.414815233 + 1.409112616i\)
\(L(\frac12)\) \(\approx\) \(1.414815233 + 1.409112616i\)
\(L(1)\) \(\approx\) \(1.128044375 + 0.4027039647i\)
\(L(1)\) \(\approx\) \(1.128044375 + 0.4027039647i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + (-0.345 + 0.938i)T \)
7 \( 1 + (0.683 + 0.730i)T \)
13 \( 1 + (0.953 - 0.299i)T \)
17 \( 1 + (0.0285 + 0.999i)T \)
19 \( 1 + (-0.610 + 0.791i)T \)
23 \( 1 + (0.981 - 0.189i)T \)
29 \( 1 + (0.179 + 0.983i)T \)
31 \( 1 + (0.398 - 0.917i)T \)
37 \( 1 + (0.198 - 0.980i)T \)
41 \( 1 + (0.625 + 0.780i)T \)
43 \( 1 + (0.786 - 0.618i)T \)
47 \( 1 + (0.548 + 0.836i)T \)
53 \( 1 + (-0.516 - 0.856i)T \)
59 \( 1 + (0.988 + 0.151i)T \)
61 \( 1 + (0.964 - 0.263i)T \)
67 \( 1 + (-0.0475 - 0.998i)T \)
71 \( 1 + (-0.564 - 0.825i)T \)
73 \( 1 + (0.0855 + 0.996i)T \)
79 \( 1 + (0.0665 + 0.997i)T \)
83 \( 1 + (0.123 + 0.992i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (0.345 + 0.938i)T \)
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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

−17.98272568240855811291491531857, −17.379812069719182214676373990314, −16.90423850991406819797800562846, −16.02707991994695369655466440075, −15.67988610913581522786189449975, −14.77623795847254799547821716836, −13.954672999769587445640700180161, −13.38067053277151122778169550413, −12.90085171125887637159043254634, −11.80150438607920739450868322954, −11.47132142681334806592247109319, −10.73161413058731295207779253255, −9.92332567105304164050371600249, −8.86375437228086234372648369443, −8.72430333338888816843548663210, −7.71860749053949371329553779745, −7.174131370629366889416872242145, −6.30063206646705654419756085777, −5.328966104564381286576531641852, −4.63590108920545910847324453822, −4.205333717031831464298318848583, −3.26966962047478114982170894447, −2.23704886585474847754526429745, −1.17699603355492293250336159435, −0.69672309758901310289622971709, 1.03007073862384129963533589395, 2.01363389898028999145251919799, 2.69228169705525387263619839036, 3.6319349984921124550140524616, 4.17265979275610577422594345788, 5.25069686147425439789387354766, 6.014413183541674888856544040238, 6.49279926531508244168642590267, 7.50393335831092526179412103435, 8.1471745690249378306451387554, 8.66605339768719818971373424473, 9.55029818116238823022167446515, 10.5890868672139120189996408334, 10.910608386527151050171240438269, 11.5024003701896009847961648091, 12.452637002157074110050667998277, 12.894252759374426444495464364989, 13.93757273971568229656232974177, 14.65470421483213279426620345785, 14.94926152027427808959542317780, 15.68754896329696779352489744446, 16.34899127207368686937648489789, 17.34754671171801489121425307043, 17.845464659456902420397489858256, 18.585855116320574355805356852533

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