Properties

Label 1-3381-3381.1559-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.670 - 0.741i$
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  + (0.476 − 0.879i)2-s + (−0.546 − 0.837i)4-s + (0.894 − 0.446i)5-s + (−0.996 + 0.0815i)8-s + (0.0339 − 0.999i)10-s + (−0.390 + 0.920i)11-s + (−0.262 − 0.965i)13-s + (−0.403 + 0.915i)16-s + (0.998 + 0.0543i)17-s + (−0.580 + 0.814i)19-s + (−0.862 − 0.505i)20-s + (0.623 + 0.781i)22-s + (0.601 − 0.798i)25-s + (−0.973 − 0.229i)26-s + (0.452 + 0.891i)29-s + ⋯
L(s)  = 1  + (0.476 − 0.879i)2-s + (−0.546 − 0.837i)4-s + (0.894 − 0.446i)5-s + (−0.996 + 0.0815i)8-s + (0.0339 − 0.999i)10-s + (−0.390 + 0.920i)11-s + (−0.262 − 0.965i)13-s + (−0.403 + 0.915i)16-s + (0.998 + 0.0543i)17-s + (−0.580 + 0.814i)19-s + (−0.862 − 0.505i)20-s + (0.623 + 0.781i)22-s + (0.601 − 0.798i)25-s + (−0.973 − 0.229i)26-s + (0.452 + 0.891i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.670 - 0.741i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.670 - 0.741i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.115612197 - 0.9392860028i\)
\(L(\frac12)\) \(\approx\) \(2.115612197 - 0.9392860028i\)
\(L(1)\) \(\approx\) \(1.290028871 - 0.6501489906i\)
\(L(1)\) \(\approx\) \(1.290028871 - 0.6501489906i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.476 - 0.879i)T \)
5 \( 1 + (0.894 - 0.446i)T \)
11 \( 1 + (-0.390 + 0.920i)T \)
13 \( 1 + (-0.262 - 0.965i)T \)
17 \( 1 + (0.998 + 0.0543i)T \)
19 \( 1 + (-0.580 + 0.814i)T \)
29 \( 1 + (0.452 + 0.891i)T \)
31 \( 1 + (0.786 + 0.618i)T \)
37 \( 1 + (0.990 - 0.135i)T \)
41 \( 1 + (-0.0611 + 0.998i)T \)
43 \( 1 + (0.996 + 0.0815i)T \)
47 \( 1 + (0.0747 + 0.997i)T \)
53 \( 1 + (-0.534 + 0.844i)T \)
59 \( 1 + (-0.0339 + 0.999i)T \)
61 \( 1 + (-0.288 + 0.957i)T \)
67 \( 1 + (-0.888 + 0.458i)T \)
71 \( 1 + (-0.101 - 0.994i)T \)
73 \( 1 + (-0.938 + 0.346i)T \)
79 \( 1 + (-0.327 + 0.945i)T \)
83 \( 1 + (0.742 - 0.670i)T \)
89 \( 1 + (0.440 - 0.897i)T \)
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

−18.90478237313301895534375293600, −18.0847103996263019338538522914, −17.27309768836412014737172833172, −16.92041248228325024576511705768, −16.14005708108456033590644181497, −15.44516151335651323548904362991, −14.61232094986572361024265799503, −14.11704076129243049504800471402, −13.513627335822268864160040848326, −12.99750884549550783932251795093, −12.00395907700548330092391739896, −11.341692969222819537199788481570, −10.40446953430172752640734008122, −9.57838708583000908007957491874, −8.987250776593682895176756390986, −8.13163270822639881343451304978, −7.41921453749035406415574621738, −6.51012802896550448635555701582, −6.132749293937369804974665078242, −5.33718638758243458979568858966, −4.60511232243279694132911360017, −3.677031345548185184681633810900, −2.81135522452999576760581795261, −2.14552580161008308800853305830, −0.64764388787240573109740887223, 1.01783693729340667126222592592, 1.59235162437520284679788393570, 2.63345544926015148266785482816, 3.08509282231357562598825770528, 4.390326515260204808979562878558, 4.7995287387905920383342490708, 5.77031610740885561908389152622, 6.07220305746849089540584414359, 7.358659672354146252756042192465, 8.22674379958734912147004549849, 9.062527590193455279551714531370, 9.89399148232459204610042976959, 10.22886878577644541504621561912, 10.84457104890286863947804916872, 12.07897176110609780089270146674, 12.4932618771836399078883033435, 12.95888504440134960353827578708, 13.745220911486946923674564892612, 14.539446338368346571561257333726, 14.91291824672072243066520076267, 15.92316050685094207661206888578, 16.75487815102257438749172806822, 17.675135696789405536640444358650, 17.95837961208461280780902413801, 18.77830539309131261405579384295

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