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Properties

Label 1-3648-3648.1877-r0-0-0
Degree $1$
Conductor $3648$
Sign $0.997 + 0.0649i$
Analytic cond. $16.9412$
Root an. cond. $16.9412$
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  + (0.843 + 0.537i)5-s + (0.258 − 0.965i)7-s + (0.793 − 0.608i)11-s + (0.0436 + 0.999i)13-s + (−0.642 + 0.766i)17-s + (0.819 + 0.573i)23-s + (0.422 + 0.906i)25-s + (0.461 − 0.887i)29-s + (−0.5 − 0.866i)31-s + (0.737 − 0.675i)35-s + (0.382 − 0.923i)37-s + (0.422 − 0.906i)41-s + (−0.537 + 0.843i)43-s + (0.642 + 0.766i)47-s + (−0.866 − 0.5i)49-s + ⋯
L(s)  = 1  + (0.843 + 0.537i)5-s + (0.258 − 0.965i)7-s + (0.793 − 0.608i)11-s + (0.0436 + 0.999i)13-s + (−0.642 + 0.766i)17-s + (0.819 + 0.573i)23-s + (0.422 + 0.906i)25-s + (0.461 − 0.887i)29-s + (−0.5 − 0.866i)31-s + (0.737 − 0.675i)35-s + (0.382 − 0.923i)37-s + (0.422 − 0.906i)41-s + (−0.537 + 0.843i)43-s + (0.642 + 0.766i)47-s + (−0.866 − 0.5i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $0.997 + 0.0649i$
Analytic conductor: \(16.9412\)
Root analytic conductor: \(16.9412\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (1877, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3648,\ (0:\ ),\ 0.997 + 0.0649i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.376687382 + 0.07731027764i\)
\(L(\frac12)\) \(\approx\) \(2.376687382 + 0.07731027764i\)
\(L(1)\) \(\approx\) \(1.383120966 + 0.02143980539i\)
\(L(1)\) \(\approx\) \(1.383120966 + 0.02143980539i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (0.843 + 0.537i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
11 \( 1 + (0.793 - 0.608i)T \)
13 \( 1 + (0.0436 + 0.999i)T \)
17 \( 1 + (-0.642 + 0.766i)T \)
23 \( 1 + (0.819 + 0.573i)T \)
29 \( 1 + (0.461 - 0.887i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.382 - 0.923i)T \)
41 \( 1 + (0.422 - 0.906i)T \)
43 \( 1 + (-0.537 + 0.843i)T \)
47 \( 1 + (0.642 + 0.766i)T \)
53 \( 1 + (-0.216 + 0.976i)T \)
59 \( 1 + (0.887 - 0.461i)T \)
61 \( 1 + (-0.537 - 0.843i)T \)
67 \( 1 + (0.461 - 0.887i)T \)
71 \( 1 + (0.819 - 0.573i)T \)
73 \( 1 + (-0.422 + 0.906i)T \)
79 \( 1 + (0.342 + 0.939i)T \)
83 \( 1 + (0.608 - 0.793i)T \)
89 \( 1 + (0.422 + 0.906i)T \)
97 \( 1 + (0.766 + 0.642i)T \)
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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

−18.383158609690218534680479098967, −17.957748795521551567541701077768, −17.453928980486914109760886507649, −16.61073620306057395519432706470, −15.99374324548914547301954030323, −15.05011156080889831422712993931, −14.70636549431429438222646657141, −13.79257889089210970941827632331, −13.05863409595853364904341633243, −12.4730273989863179832783453089, −11.87950010517583335922051139073, −11.02150049343222758711801933255, −10.16079855443524176020413527262, −9.52283907771693952866778595475, −8.71636787614350136599889691464, −8.498240684094937065195110790800, −7.19232042300654036593181900548, −6.57326685147488237418492331353, −5.715584823175115627667078222147, −5.04196109272133698414676394472, −4.570551101536692309248503311701, −3.24682414965292850577384595091, −2.51899640098693032513428706610, −1.72874865021329279237784758393, −0.873873046519164033742365323927, 0.8961088094394622261865254206, 1.73052775415268809046148748537, 2.473533161202928779889941977235, 3.64410399969471568601991750953, 4.093865470826625170007107094415, 5.053664311757022400892164199451, 6.11879736494548883842175879856, 6.47803027862385836495439395573, 7.26120318088439292942776713, 8.03115576808259378908494953496, 9.16685548626111840640546498657, 9.39763942412624274981648056351, 10.42106112553940147807529809715, 11.09584951504733646627099200974, 11.39903420413398845492263504222, 12.58020402399793161058084816190, 13.38618092867937553464259591160, 13.877392979891249138986305842856, 14.39726516742546532950019838882, 15.072525717040870763000383025960, 16.030544050348943094999055046903, 16.95616385702919224706503813766, 17.15393428065523367065629645623, 17.82953881495237420151088679601, 18.79081751761184620351027001071

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