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Properties

Label 1-1476-1476.419-r0-0-0
Degree $1$
Conductor $1476$
Sign $0.999 + 0.0116i$
Analytic cond. $6.85451$
Root an. cond. $6.85451$
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  + (−0.5 − 0.866i)5-s + (0.866 + 0.5i)7-s + (−0.866 − 0.5i)11-s + (0.866 − 0.5i)13-s + i·17-s − i·19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.866 + 0.5i)29-s + (0.5 + 0.866i)31-s − i·35-s + 37-s + (−0.5 + 0.866i)43-s + (0.866 + 0.5i)47-s + (0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + (0.866 + 0.5i)7-s + (−0.866 − 0.5i)11-s + (0.866 − 0.5i)13-s + i·17-s − i·19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.866 + 0.5i)29-s + (0.5 + 0.866i)31-s − i·35-s + 37-s + (−0.5 + 0.866i)43-s + (0.866 + 0.5i)47-s + (0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1476\)    =    \(2^{2} \cdot 3^{2} \cdot 41\)
Sign: $0.999 + 0.0116i$
Analytic conductor: \(6.85451\)
Root analytic conductor: \(6.85451\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1476} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1476,\ (0:\ ),\ 0.999 + 0.0116i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.524528426 + 0.008915070708i\)
\(L(\frac12)\) \(\approx\) \(1.524528426 + 0.008915070708i\)
\(L(1)\) \(\approx\) \(1.088125857 - 0.05973556674i\)
\(L(1)\) \(\approx\) \(1.088125857 - 0.05973556674i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
41 \( 1 \)
good5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (0.866 - 0.5i)T \)
17 \( 1 + iT \)
19 \( 1 - iT \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.866 + 0.5i)T \)
53 \( 1 - iT \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 - iT \)
73 \( 1 - T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 - iT \)
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

−20.6034702317715111667709939617, −20.02412775786700582447021163505, −19.08678308351879555992561805628, −18.258978094320552657251036856126, −17.93154157307534376897364370541, −17.011152139197316748453975134843, −15.89032769803290071861378759984, −15.46621780782295086156272007043, −14.67918966319279826774383766575, −13.62675357275617330200429336404, −13.51895623557508528722192059585, −11.92066146253930604942680870418, −11.52828377149508128141331464617, −10.74179161252501526940986590838, −10.08835075633755746126621335867, −9.03970149576578932006890814450, −7.99835484799663132629461271537, −7.46790641270308477139138243691, −6.75513078217155185984565687412, −5.69782919836625315669816866287, −4.629372442104017358433903771006, −4.04048669618211818763701800665, −2.881661366230208676846413056423, −2.12881445102279106368826795611, −0.76628871681462130854248440382, 0.91905318801310440433436077276, 1.80916178258967241086655799934, 3.01775936785497114573886316457, 4.00360651757603379130307363616, 4.851305085521413595798958641787, 5.60571214680886951700379912716, 6.35271101411980483902152366084, 7.86599949782615984380453320993, 8.273143229878319200905976872430, 8.65480357968669783390590159770, 9.97139782929160309687322384623, 10.79450912781933623796428477327, 11.452940432198992023996700455761, 12.496199734398336627662486207379, 12.770547779605824322846704683876, 13.88566292547907769001361582197, 14.636925030369792230064939824636, 15.57426898341841564904987790380, 16.037481428342000517520598621645, 16.83016525689372503719004797183, 17.76471644120735762028884478099, 18.391897197831519180130021567342, 19.113708718298117302313469444775, 20.07137442376104012352410799541, 20.71143739096206386153441691092

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