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Properties

Label 1-3381-3381.173-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.632 - 0.774i$
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  + (0.546 − 0.837i)2-s + (−0.403 − 0.915i)4-s + (0.601 + 0.798i)5-s + (−0.986 − 0.162i)8-s + (0.997 − 0.0679i)10-s + (0.694 − 0.719i)11-s + (0.862 + 0.505i)13-s + (−0.675 + 0.737i)16-s + (0.994 − 0.108i)17-s + (0.327 − 0.945i)19-s + (0.488 − 0.872i)20-s + (−0.222 − 0.974i)22-s + (−0.275 + 0.961i)25-s + (0.894 − 0.446i)26-s + (0.591 + 0.806i)29-s + ⋯
L(s)  = 1  + (0.546 − 0.837i)2-s + (−0.403 − 0.915i)4-s + (0.601 + 0.798i)5-s + (−0.986 − 0.162i)8-s + (0.997 − 0.0679i)10-s + (0.694 − 0.719i)11-s + (0.862 + 0.505i)13-s + (−0.675 + 0.737i)16-s + (0.994 − 0.108i)17-s + (0.327 − 0.945i)19-s + (0.488 − 0.872i)20-s + (−0.222 − 0.974i)22-s + (−0.275 + 0.961i)25-s + (0.894 − 0.446i)26-s + (0.591 + 0.806i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.632 - 0.774i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.632 - 0.774i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.621267463 - 1.244619942i\)
\(L(\frac12)\) \(\approx\) \(2.621267463 - 1.244619942i\)
\(L(1)\) \(\approx\) \(1.547687275 - 0.6100528235i\)
\(L(1)\) \(\approx\) \(1.547687275 - 0.6100528235i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.546 - 0.837i)T \)
5 \( 1 + (0.601 + 0.798i)T \)
11 \( 1 + (0.694 - 0.719i)T \)
13 \( 1 + (0.862 + 0.505i)T \)
17 \( 1 + (0.994 - 0.108i)T \)
19 \( 1 + (0.327 - 0.945i)T \)
29 \( 1 + (0.591 + 0.806i)T \)
31 \( 1 + (-0.235 + 0.971i)T \)
37 \( 1 + (0.963 + 0.268i)T \)
41 \( 1 + (-0.992 + 0.122i)T \)
43 \( 1 + (0.986 - 0.162i)T \)
47 \( 1 + (-0.988 - 0.149i)T \)
53 \( 1 + (0.427 - 0.903i)T \)
59 \( 1 + (-0.997 + 0.0679i)T \)
61 \( 1 + (0.833 - 0.552i)T \)
67 \( 1 + (0.580 + 0.814i)T \)
71 \( 1 + (0.979 + 0.202i)T \)
73 \( 1 + (-0.760 - 0.649i)T \)
79 \( 1 + (-0.786 + 0.618i)T \)
83 \( 1 + (0.101 + 0.994i)T \)
89 \( 1 + (-0.612 + 0.790i)T \)
97 \( 1 + (-0.841 + 0.540i)T \)
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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

−18.5763679574254098632738692366, −18.127418819940759839470221339063, −17.09849860922997546294168587821, −17.023690772719709203819371204336, −16.11260880615928778860883128365, −15.558674864275874812375704150861, −14.62564153797449178063669057683, −14.18943887673289959801162714810, −13.35805009377389501296313590216, −12.830532029262236942311658315264, −12.148628057427326093117265739655, −11.56988372380731933168858931091, −10.25296905532108184432273202872, −9.608647046871523270077497012857, −8.946870796366440985576941160717, −8.06247783552800784631421028130, −7.66725613034632371278277803107, −6.51387453120788111595263290830, −5.91794194759265156912365607823, −5.420468880818201327912047868895, −4.43769862160916448048240599245, −3.90882772128642816867387708947, −2.954444086666992532348364873794, −1.81775248279358877240985009369, −0.88404613416924359266811313607, 0.9964540686029528712455672078, 1.607791849156449679086487172513, 2.70075884187196266014798515694, 3.28569808522054193792630381853, 3.91183955381965336621951078419, 5.02528721096957762019070525870, 5.662636624931543943854871899720, 6.504750346867583810774776765308, 6.91111832660775648717265864855, 8.30217156360967784361341862963, 9.065807956232227330117434162957, 9.70447794074302528110082799586, 10.42353979664294506394464550789, 11.20461309812143197157319180620, 11.49798458913024205255147369249, 12.44611576692290229115070703390, 13.261031937851253528039709153777, 13.94706099374693828782524221133, 14.251488766488429797007941034320, 14.976895869044648429948483594812, 15.885005114792074271351105265027, 16.62176482932262063023934055028, 17.62115301227788259607641593167, 18.2325512895068449226294814100, 18.78646439067388248990131744554

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