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Properties

Label 1-3381-3381.248-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.426 + 0.904i$
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  + (0.248 + 0.968i)2-s + (−0.876 + 0.482i)4-s + (−0.0882 − 0.996i)5-s + (−0.685 − 0.728i)8-s + (0.942 − 0.333i)10-s + (−0.644 + 0.764i)11-s + (−0.882 + 0.470i)13-s + (0.534 − 0.844i)16-s + (0.855 − 0.517i)17-s + (0.995 + 0.0950i)19-s + (0.557 + 0.830i)20-s + (−0.900 − 0.433i)22-s + (−0.984 + 0.175i)25-s + (−0.675 − 0.737i)26-s + (−0.0203 − 0.999i)29-s + ⋯
L(s)  = 1  + (0.248 + 0.968i)2-s + (−0.876 + 0.482i)4-s + (−0.0882 − 0.996i)5-s + (−0.685 − 0.728i)8-s + (0.942 − 0.333i)10-s + (−0.644 + 0.764i)11-s + (−0.882 + 0.470i)13-s + (0.534 − 0.844i)16-s + (0.855 − 0.517i)17-s + (0.995 + 0.0950i)19-s + (0.557 + 0.830i)20-s + (−0.900 − 0.433i)22-s + (−0.984 + 0.175i)25-s + (−0.675 − 0.737i)26-s + (−0.0203 − 0.999i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.426 + 0.904i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (248, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.426 + 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.105398873 + 0.7011379953i\)
\(L(\frac12)\) \(\approx\) \(1.105398873 + 0.7011379953i\)
\(L(1)\) \(\approx\) \(0.9012314152 + 0.3744611733i\)
\(L(1)\) \(\approx\) \(0.9012314152 + 0.3744611733i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.248 + 0.968i)T \)
5 \( 1 + (-0.0882 - 0.996i)T \)
11 \( 1 + (-0.644 + 0.764i)T \)
13 \( 1 + (-0.882 + 0.470i)T \)
17 \( 1 + (0.855 - 0.517i)T \)
19 \( 1 + (0.995 + 0.0950i)T \)
29 \( 1 + (-0.0203 - 0.999i)T \)
31 \( 1 + (-0.928 + 0.371i)T \)
37 \( 1 + (0.209 + 0.977i)T \)
41 \( 1 + (-0.818 + 0.574i)T \)
43 \( 1 + (0.685 - 0.728i)T \)
47 \( 1 + (-0.733 - 0.680i)T \)
53 \( 1 + (0.802 + 0.596i)T \)
59 \( 1 + (-0.942 + 0.333i)T \)
61 \( 1 + (-0.976 - 0.215i)T \)
67 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (0.523 + 0.852i)T \)
73 \( 1 + (0.923 + 0.384i)T \)
79 \( 1 + (0.981 - 0.189i)T \)
83 \( 1 + (0.488 + 0.872i)T \)
89 \( 1 + (0.155 - 0.987i)T \)
97 \( 1 + (0.959 + 0.281i)T \)
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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

−18.68164208074701345741088699599, −18.25744356304238419176863974445, −17.646206538139032947694061838108, −16.69114337227441701277145560072, −15.82767986481367981658293561459, −14.92168582917332501702977987566, −14.46951908631140783528025270270, −13.84661888417632241295919035427, −13.043609395834593404906037033775, −12.36425908153419135435936558887, −11.66940601760873598685596803735, −10.8450069289628812655188949447, −10.52553666780865167038609255235, −9.73458833329915128912939431937, −9.06037930274815844365126243213, −7.905824776758799176451307983210, −7.53002851859553299772984248908, −6.33297447278836393286082948096, −5.52644934737592505993626578184, −5.03209392007539792178858942837, −3.769130716777525655970163786370, −3.256760695116295835549320549630, −2.645824045817280422571177187231, −1.75420121992227881089168897745, −0.57349859481517161586280904520, 0.64121456173089846078451750995, 1.822049087657405059602989186876, 2.97099664678536468873712243383, 3.89976223167724858837179523522, 4.84273323253895979348123306719, 5.08234960725109324461166306447, 5.88290802968041798348188161455, 6.94292154789967942072314389729, 7.61448972715149842316060987169, 8.02857153195490517991160464896, 9.04037804689282957524567270602, 9.624821441439833583462635013447, 10.12155387815849950236982021095, 11.613438078493148194392209519996, 12.20793279386368275517666723876, 12.667543616841126065422777697553, 13.60109073918473797332869878500, 13.98727444692965644259416854857, 15.02135019964215378476776504045, 15.44123334250089078778023439855, 16.281181111747694715126870769971, 16.77561630312879189131802495812, 17.28166920936522140088272945854, 18.2156673439483552981918358577, 18.62716057043404254612687552275

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