Properties

Label 1-3381-3381.1331-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.986 - 0.163i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (0.0611 + 0.998i)2-s + (−0.992 + 0.122i)4-s + (−0.862 − 0.505i)5-s + (−0.182 − 0.983i)8-s + (0.452 − 0.891i)10-s + (0.488 − 0.872i)11-s + (−0.979 + 0.202i)13-s + (0.970 − 0.242i)16-s + (−0.992 − 0.122i)17-s + (−0.841 + 0.540i)19-s + (0.917 + 0.396i)20-s + (0.900 + 0.433i)22-s + (0.488 + 0.872i)25-s + (−0.262 − 0.965i)26-s + (0.992 + 0.122i)29-s + ⋯
L(s)  = 1  + (0.0611 + 0.998i)2-s + (−0.992 + 0.122i)4-s + (−0.862 − 0.505i)5-s + (−0.182 − 0.983i)8-s + (0.452 − 0.891i)10-s + (0.488 − 0.872i)11-s + (−0.979 + 0.202i)13-s + (0.970 − 0.242i)16-s + (−0.992 − 0.122i)17-s + (−0.841 + 0.540i)19-s + (0.917 + 0.396i)20-s + (0.900 + 0.433i)22-s + (0.488 + 0.872i)25-s + (−0.262 − 0.965i)26-s + (0.992 + 0.122i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.986 - 0.163i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.986 - 0.163i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6941966014 - 0.05707490533i\)
\(L(\frac12)\) \(\approx\) \(0.6941966014 - 0.05707490533i\)
\(L(1)\) \(\approx\) \(0.6779981898 + 0.2433656034i\)
\(L(1)\) \(\approx\) \(0.6779981898 + 0.2433656034i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.0611 + 0.998i)T \)
5 \( 1 + (-0.862 - 0.505i)T \)
11 \( 1 + (0.488 - 0.872i)T \)
13 \( 1 + (-0.979 + 0.202i)T \)
17 \( 1 + (-0.992 - 0.122i)T \)
19 \( 1 + (-0.841 + 0.540i)T \)
29 \( 1 + (0.992 + 0.122i)T \)
31 \( 1 + (-0.654 + 0.755i)T \)
37 \( 1 + (0.301 + 0.953i)T \)
41 \( 1 + (0.862 + 0.505i)T \)
43 \( 1 + (-0.182 + 0.983i)T \)
47 \( 1 + (0.222 - 0.974i)T \)
53 \( 1 + (-0.768 + 0.639i)T \)
59 \( 1 + (0.452 - 0.891i)T \)
61 \( 1 + (-0.262 + 0.965i)T \)
67 \( 1 + (0.959 - 0.281i)T \)
71 \( 1 + (-0.986 - 0.162i)T \)
73 \( 1 + (-0.714 - 0.699i)T \)
79 \( 1 + (-0.415 - 0.909i)T \)
83 \( 1 + (0.996 - 0.0815i)T \)
89 \( 1 + (-0.591 + 0.806i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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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

−19.12786077802701188266695681374, −18.24664665044648559109699559990, −17.521949964810932853047313406853, −17.15938832617435367641563602555, −15.91614097605060584237980532733, −15.19447005635237535950882718392, −14.62673901009821091039141036206, −14.05852151386939696448463207786, −12.89515235559951019765862161630, −12.57389568169053937635946443490, −11.79577451416265188744338192014, −11.16233307080119639581916973757, −10.57558542302476781712151110587, −9.80514845503650701235768283904, −9.07801782239249321378679256681, −8.3511883129600578411313924925, −7.44376667855458964536555632785, −6.84454731588466740105814809344, −5.80010361889612833830564474787, −4.53867608613094809231869092840, −4.41991849373865910870062568834, −3.49272568055244451628837790041, −2.45642161528186997873599820574, −2.11270773781190308208931584807, −0.67399663269001217156360248680, 0.320968651327075440464607376, 1.42165511445776219199458584818, 2.85876196190126313389694325792, 3.7343990812949569295270735826, 4.522563396031469194487646720527, 4.92787350529175430287634022706, 6.0304846755199540811966339970, 6.64200374142062025547307622175, 7.40616033340794651271226899069, 8.17220288330091549186746966270, 8.711481551912714990007775187938, 9.29871494526461920794548035300, 10.257435346584169406054776646030, 11.19196106694719353347385175397, 11.978571507340592058411038313001, 12.62972630797364060366763578311, 13.29837541989346081038425150965, 14.14629169668847125265799069160, 14.7637944895092857563300005359, 15.350639832895992886452768489796, 16.2281409345664679155842838830, 16.50455838396925288207574230003, 17.250051035652432100185428020, 17.92462679537104465749008769846, 18.84392795376329039962061224894

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