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Properties

Label 1-3381-3381.1394-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.925 + 0.377i$
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  + (−0.101 + 0.994i)2-s + (−0.979 − 0.202i)4-s + (0.986 + 0.162i)5-s + (0.301 − 0.953i)8-s + (−0.262 + 0.965i)10-s + (0.947 − 0.320i)11-s + (0.182 + 0.983i)13-s + (0.917 + 0.396i)16-s + (−0.979 + 0.202i)17-s + (−0.415 + 0.909i)19-s + (−0.933 − 0.359i)20-s + (0.222 + 0.974i)22-s + (0.947 + 0.320i)25-s + (−0.996 + 0.0815i)26-s + (0.979 − 0.202i)29-s + ⋯
L(s)  = 1  + (−0.101 + 0.994i)2-s + (−0.979 − 0.202i)4-s + (0.986 + 0.162i)5-s + (0.301 − 0.953i)8-s + (−0.262 + 0.965i)10-s + (0.947 − 0.320i)11-s + (0.182 + 0.983i)13-s + (0.917 + 0.396i)16-s + (−0.979 + 0.202i)17-s + (−0.415 + 0.909i)19-s + (−0.933 − 0.359i)20-s + (0.222 + 0.974i)22-s + (0.947 + 0.320i)25-s + (−0.996 + 0.0815i)26-s + (0.979 − 0.202i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.925 + 0.377i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1394, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.925 + 0.377i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2895275731 + 1.475485274i\)
\(L(\frac12)\) \(\approx\) \(0.2895275731 + 1.475485274i\)
\(L(1)\) \(\approx\) \(0.8596241557 + 0.6474365827i\)
\(L(1)\) \(\approx\) \(0.8596241557 + 0.6474365827i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.101 + 0.994i)T \)
5 \( 1 + (0.986 + 0.162i)T \)
11 \( 1 + (0.947 - 0.320i)T \)
13 \( 1 + (0.182 + 0.983i)T \)
17 \( 1 + (-0.979 + 0.202i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (0.979 - 0.202i)T \)
31 \( 1 + (-0.142 - 0.989i)T \)
37 \( 1 + (-0.488 + 0.872i)T \)
41 \( 1 + (-0.986 - 0.162i)T \)
43 \( 1 + (0.301 + 0.953i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
53 \( 1 + (-0.591 + 0.806i)T \)
59 \( 1 + (-0.262 + 0.965i)T \)
61 \( 1 + (-0.996 - 0.0815i)T \)
67 \( 1 + (-0.841 + 0.540i)T \)
71 \( 1 + (0.714 + 0.699i)T \)
73 \( 1 + (0.970 - 0.242i)T \)
79 \( 1 + (0.654 - 0.755i)T \)
83 \( 1 + (-0.377 - 0.925i)T \)
89 \( 1 + (-0.862 + 0.505i)T \)
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

−18.31449323391430814118216416848, −17.92422597306659239578708689169, −17.36126058004363967270804789508, −16.82880600722084955934865993701, −15.70142911528001339179331269317, −14.92161617095430923179439890243, −13.98976578959564840242091417107, −13.67364636072541622866364810919, −12.762029420454582001804319814548, −12.40598564670347923730127827879, −11.43211728974940098342134816303, −10.705644942935724909962700199190, −10.20455158998413476943721480464, −9.33270522792164908931951642001, −8.88376535617840277140560914374, −8.217610814208334734412702884907, −6.94421443865575460373370922052, −6.352029595072342388871729696474, −5.19205435005881763433842473302, −4.83343748819939640748765790578, −3.78431754683374207763088370260, −2.97436706864047492077136314853, −2.143636920818298574656436886180, −1.49215230001119342306721500564, −0.467826095120685012376002816648, 1.22258456162855547342393617812, 1.87552846767518392761432651016, 3.11783185160993094450451272595, 4.17756060366346441028511841832, 4.6595007192992525711579013013, 5.763118740343128300570543878557, 6.458202669100024035260597898566, 6.57187630142318336132851479872, 7.71698075996631296855574883389, 8.62397938443379051860323595615, 9.09730977298576950341992285019, 9.76984768069045128531171893451, 10.48609913081444580307077964383, 11.39118483503605766396170639689, 12.307050304040885327172192809954, 13.176569766593212144240413368771, 13.81494568601782540690627030152, 14.23055033859633959060627010114, 14.93124473812898815922685048137, 15.666144267051947850940250372157, 16.69297391216488241209837255568, 16.84069210829826705675636163591, 17.61977268749081589956167626736, 18.27033883315947715375941784441, 18.96375915559578996152247522240

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