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Properties

Label 1-4015-4015.2793-r0-0-0
Degree $1$
Conductor $4015$
Sign $-0.313 + 0.949i$
Analytic cond. $18.6455$
Root an. cond. $18.6455$
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  + (0.642 + 0.766i)2-s + (−0.866 + 0.5i)3-s + (−0.173 + 0.984i)4-s + (−0.939 − 0.342i)6-s + (−0.5 − 0.866i)7-s + (−0.866 + 0.5i)8-s + (0.5 − 0.866i)9-s + (−0.342 − 0.939i)12-s + (−0.939 − 0.342i)13-s + (0.342 − 0.939i)14-s + (−0.939 − 0.342i)16-s + (−0.5 + 0.866i)17-s + (0.984 − 0.173i)18-s + (−0.766 + 0.642i)19-s + (0.866 + 0.5i)21-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (−0.866 + 0.5i)3-s + (−0.173 + 0.984i)4-s + (−0.939 − 0.342i)6-s + (−0.5 − 0.866i)7-s + (−0.866 + 0.5i)8-s + (0.5 − 0.866i)9-s + (−0.342 − 0.939i)12-s + (−0.939 − 0.342i)13-s + (0.342 − 0.939i)14-s + (−0.939 − 0.342i)16-s + (−0.5 + 0.866i)17-s + (0.984 − 0.173i)18-s + (−0.766 + 0.642i)19-s + (0.866 + 0.5i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4015\)    =    \(5 \cdot 11 \cdot 73\)
Sign: $-0.313 + 0.949i$
Analytic conductor: \(18.6455\)
Root analytic conductor: \(18.6455\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4015} (2793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4015,\ (0:\ ),\ -0.313 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5866658299 + 0.8116801206i\)
\(L(\frac12)\) \(\approx\) \(0.5866658299 + 0.8116801206i\)
\(L(1)\) \(\approx\) \(0.7413720774 + 0.4579748965i\)
\(L(1)\) \(\approx\) \(0.7413720774 + 0.4579748965i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
73 \( 1 \)
good2 \( 1 + (0.642 + 0.766i)T \)
3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (0.642 - 0.766i)T \)
29 \( 1 + (0.642 + 0.766i)T \)
31 \( 1 + (-0.984 + 0.173i)T \)
37 \( 1 + (0.642 - 0.766i)T \)
41 \( 1 + (0.939 - 0.342i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.939 + 0.342i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (0.342 - 0.939i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (0.939 + 0.342i)T \)
83 \( 1 - T \)
89 \( 1 + (0.939 + 0.342i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−18.26084435934205128167086587254, −17.89392431460112748749373086087, −16.94339617973634698067488908066, −16.22071628773837781594438366606, −15.39703725697316413272131442117, −14.922905115974299368442521250116, −13.94412561432784414608983452219, −13.12546170106104477113440940901, −12.87101518958828050421709924139, −11.99929581891263658410993618225, −11.5445461689456914496572809774, −11.04843841921672871482087023583, −10.02858537961969570693547915386, −9.50848910081900957346932306577, −8.7677454951466608306074075483, −7.5554674849704745339322732630, −6.719887619409561245727491031761, −6.23978591378779558200281930100, −5.37033879542733628188220189971, −4.85524778283760547416943145854, −4.14441480768816963120015355084, −2.809444960276178274750029001315, −2.4639846856041829740790105588, −1.535407612859445886178359920366, −0.431517514262311558606770526284, 0.57388730561134009710380664050, 2.07807478572770551173438098274, 3.26475991717498355163089156518, 3.8898065337784667454270059921, 4.55732677037132943879586220000, 5.18429369661728182643323146363, 6.03739686042558409810261702376, 6.6446641365525984418389076934, 7.17139292590623833926960211147, 8.043043519513749717508933021660, 8.94661764404519659945596973238, 9.71960793744900190773452177766, 10.619875514475027548690378227577, 10.94483479630633849598346860821, 12.08167251120195614883475173079, 12.80324114477505965139079775788, 12.86996310727744911129130595100, 14.125106788462448845223376240675, 14.71544406432565910719622312715, 15.250724916155212130854067358647, 16.106031209283843119705140152831, 16.6645201778859536292289019531, 17.020497998013000965180636356115, 17.6964443134986696976874150752, 18.3456200956343674665588342165

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