↑

Properties

Label 2-381-127.108-c2-0-10
Degree $2$
Conductor $381$
Sign $0.163 - 0.986i$
Analytic cond. $10.3814$
Root an. cond. $3.22203$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.06·2-s + (−1.5 − 0.866i)3-s + 5.36·4-s + 4.88i·5-s + (4.59 + 2.65i)6-s + (8.45 + 4.87i)7-s − 4.18·8-s + (1.5 + 2.59i)9-s − 14.9i·10-s + (7.89 + 13.6i)11-s + (−8.04 − 4.64i)12-s + (6.97 − 12.0i)13-s + (−25.8 − 14.9i)14-s + (4.23 − 7.32i)15-s − 8.66·16-s + (5.99 + 10.3i)17-s + ⋯
L(s)  = 1  − 1.53·2-s + (−0.5 − 0.288i)3-s + 1.34·4-s + 0.977i·5-s + (0.765 + 0.441i)6-s + (1.20 + 0.697i)7-s − 0.522·8-s + (0.166 + 0.288i)9-s − 1.49i·10-s + (0.717 + 1.24i)11-s + (−0.670 − 0.387i)12-s + (0.536 − 0.928i)13-s + (−1.84 − 1.06i)14-s + (0.282 − 0.488i)15-s − 0.541·16-s + (0.352 + 0.611i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(381\)    =    \(3 \cdot 127\)
Sign: $0.163 - 0.986i$
Analytic conductor: \(10.3814\)
Root analytic conductor: \(3.22203\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{381} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 381,\ (\ :1),\ 0.163 - 0.986i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.584127 + 0.495283i\)
\(L(\frac12)\) \(\approx\) \(0.584127 + 0.495283i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 + 0.866i)T \)
127 \( 1 + (79.7 + 98.8i)T \)
good2 \( 1 + 3.06T + 4T^{2} \)
5 \( 1 - 4.88iT - 25T^{2} \)
7 \( 1 + (-8.45 - 4.87i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-7.89 - 13.6i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-6.97 + 12.0i)T + (-84.5 - 146. i)T^{2} \)
17 \( 1 + (-5.99 - 10.3i)T + (-144.5 + 250. i)T^{2} \)
19 \( 1 + 4.06T + 361T^{2} \)
23 \( 1 + (17.1 + 9.90i)T + (264.5 + 458. i)T^{2} \)
29 \( 1 + (3.39 - 1.96i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-21.9 + 38.0i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-22.2 - 38.5i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-8.11 - 14.0i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (21.6 - 12.4i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 - 41.8T + 2.20e3T^{2} \)
53 \( 1 + (49.3 - 28.5i)T + (1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (8.26 - 4.77i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + 18.4T + 3.72e3T^{2} \)
67 \( 1 + (-101. - 58.5i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + (1.53 - 2.66i)T + (-2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 - 30.0T + 5.32e3T^{2} \)
79 \( 1 + (9.85 - 17.0i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (111. - 64.3i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 63.9iT - 7.92e3T^{2} \)
97 \( 1 + (64.9 + 37.4i)T + (4.70e3 + 8.14e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.14920619851725366035614255665, −10.36080582732391002129800641138, −9.653961918790585473466617332695, −8.357179639829357843449351285897, −7.87073744160321341274828653910, −6.86415169507535922833748280550, −5.95797504421889497914473827936, −4.46725604847651085289912028091, −2.41287575969425235674358414525, −1.33783789590087295029468539602, 0.72912516999349461409788437037, 1.53871104524459918115943626235, 4.00742484124717468044187272828, 5.02613415782671168557499661651, 6.36188059467743580811538953189, 7.50364490073974802087210791650, 8.470009373541769574227016695304, 8.936926250353477848408641433608, 9.882755540593934792045117284424, 11.02446002547020250503829508856

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