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Properties

Label 1-3381-3381.428-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.949 + 0.312i$
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  + (0.523 − 0.852i)2-s + (−0.452 − 0.891i)4-s + (−0.0611 − 0.998i)5-s + (−0.996 − 0.0815i)8-s + (−0.882 − 0.470i)10-s + (−0.992 − 0.122i)11-s + (0.262 − 0.965i)13-s + (−0.591 + 0.806i)16-s + (−0.452 + 0.891i)17-s + (−0.415 + 0.909i)19-s + (−0.862 + 0.505i)20-s + (−0.623 + 0.781i)22-s + (−0.992 + 0.122i)25-s + (−0.685 − 0.728i)26-s + (0.452 − 0.891i)29-s + ⋯
L(s)  = 1  + (0.523 − 0.852i)2-s + (−0.452 − 0.891i)4-s + (−0.0611 − 0.998i)5-s + (−0.996 − 0.0815i)8-s + (−0.882 − 0.470i)10-s + (−0.992 − 0.122i)11-s + (0.262 − 0.965i)13-s + (−0.591 + 0.806i)16-s + (−0.452 + 0.891i)17-s + (−0.415 + 0.909i)19-s + (−0.862 + 0.505i)20-s + (−0.623 + 0.781i)22-s + (−0.992 + 0.122i)25-s + (−0.685 − 0.728i)26-s + (0.452 − 0.891i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.949 + 0.312i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.949 + 0.312i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4630381340 + 0.07418537781i\)
\(L(\frac12)\) \(\approx\) \(0.4630381340 + 0.07418537781i\)
\(L(1)\) \(\approx\) \(0.7599161502 - 0.5949416444i\)
\(L(1)\) \(\approx\) \(0.7599161502 - 0.5949416444i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.523 - 0.852i)T \)
5 \( 1 + (-0.0611 - 0.998i)T \)
11 \( 1 + (-0.992 - 0.122i)T \)
13 \( 1 + (0.262 - 0.965i)T \)
17 \( 1 + (-0.452 + 0.891i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (0.452 - 0.891i)T \)
31 \( 1 + (-0.142 - 0.989i)T \)
37 \( 1 + (0.377 + 0.925i)T \)
41 \( 1 + (0.0611 + 0.998i)T \)
43 \( 1 + (-0.996 + 0.0815i)T \)
47 \( 1 + (0.900 - 0.433i)T \)
53 \( 1 + (-0.999 + 0.0407i)T \)
59 \( 1 + (-0.882 - 0.470i)T \)
61 \( 1 + (-0.685 + 0.728i)T \)
67 \( 1 + (-0.841 + 0.540i)T \)
71 \( 1 + (-0.101 + 0.994i)T \)
73 \( 1 + (-0.768 + 0.639i)T \)
79 \( 1 + (0.654 - 0.755i)T \)
83 \( 1 + (0.742 + 0.670i)T \)
89 \( 1 + (0.557 - 0.830i)T \)
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

−18.52270397826872975958539262254, −18.02583984602328162300762547534, −17.50471154683810916517652575281, −16.46597345454803870336909517180, −15.898142532067269028584249277967, −15.37321587802058823580104702474, −14.666107985734916403471650015436, −13.85115714161125254229531245527, −13.62871030977597132162767472053, −12.61235842555966429498630785880, −11.93700216683954049057586869454, −11.02709878672977463392413140902, −10.565402112442654907853418740340, −9.333119795470514266070048377410, −8.88434283467475904094175768259, −7.83880703704858106614839622784, −7.20126670469756536388399936003, −6.70772488308317042176072878393, −5.99434176015483379723572011743, −4.99276747480632885305864927359, −4.495494645257832564387734796099, −3.43177885140026064174920094360, −2.81773853823427514732242650548, −2.00320930795054863843854304109, −0.12397728429522083775580928172, 0.96406239826109190433639760684, 1.81059974685535351195060919529, 2.66941647408240414775176279613, 3.55505990454145137350260448561, 4.35746989829238682130537444243, 4.950305322936306142176867649484, 5.85898109110499727431691970631, 6.185484233267615378600672270478, 7.82771508801188365215176410293, 8.23391593316337548034810447033, 9.01929451794084387589251260417, 10.02498150707706311385278049718, 10.34350866786930419310287523142, 11.250718155557455387196080230015, 11.96827366315675684093362711868, 12.714955697141280290826337733487, 13.18714383955275243329266224151, 13.57847269569522961147231516373, 14.74988203270416395291507243164, 15.308351079644413762350855824653, 15.9085213785658796697026828252, 16.88368954030184641992522276, 17.51903501391386596736398245731, 18.37733105505050817656386981146, 18.92207082929814981820783703627

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