↑

Properties

Label 2-1617-77.76-c1-0-8
Degree $2$
Conductor $1617$
Sign $-0.523 + 0.852i$
Analytic cond. $12.9118$
Root an. cond. $3.59330$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.98i·2-s − i·3-s − 1.92·4-s + 0.364i·5-s + 1.98·6-s + 0.145i·8-s − 9-s − 0.722·10-s + (3.27 − 0.538i)11-s + 1.92i·12-s − 5.43·13-s + 0.364·15-s − 4.14·16-s − 5.87·17-s − 1.98i·18-s − 2.22·19-s + ⋯
L(s)  = 1  + 1.40i·2-s − 0.577i·3-s − 0.963·4-s + 0.163i·5-s + 0.808·6-s + 0.0512i·8-s − 0.333·9-s − 0.228·10-s + (0.986 − 0.162i)11-s + 0.556i·12-s − 1.50·13-s + 0.0941·15-s − 1.03·16-s − 1.42·17-s − 0.467i·18-s − 0.510·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.523 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $-0.523 + 0.852i$
Analytic conductor: \(12.9118\)
Root analytic conductor: \(3.59330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (538, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :1/2),\ -0.523 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3560343888\)
\(L(\frac12)\) \(\approx\) \(0.3560343888\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 \)
11 \( 1 + (-3.27 + 0.538i)T \)
good2 \( 1 - 1.98iT - 2T^{2} \)
5 \( 1 - 0.364iT - 5T^{2} \)
13 \( 1 + 5.43T + 13T^{2} \)
17 \( 1 + 5.87T + 17T^{2} \)
19 \( 1 + 2.22T + 19T^{2} \)
23 \( 1 - 2.76T + 23T^{2} \)
29 \( 1 + 2.01iT - 29T^{2} \)
31 \( 1 - 5.77iT - 31T^{2} \)
37 \( 1 + 8.01T + 37T^{2} \)
41 \( 1 + 8.88T + 41T^{2} \)
43 \( 1 - 5.90iT - 43T^{2} \)
47 \( 1 - 6.29iT - 47T^{2} \)
53 \( 1 + 12.9T + 53T^{2} \)
59 \( 1 + 8.54iT - 59T^{2} \)
61 \( 1 - 2.70T + 61T^{2} \)
67 \( 1 + 14.6T + 67T^{2} \)
71 \( 1 - 2.64T + 71T^{2} \)
73 \( 1 - 5.82T + 73T^{2} \)
79 \( 1 - 1.83iT - 79T^{2} \)
83 \( 1 - 5.21T + 83T^{2} \)
89 \( 1 + 0.424iT - 89T^{2} \)
97 \( 1 + 10.0iT - 97T^{2} \)
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

−9.505261762687263016405826481390, −8.843376757120301980183294112215, −8.209283010240917267609273032813, −7.23203067477746783265002792367, −6.74492204422123818149764588435, −6.31543012954401263105937124531, −5.07542060876735976832653955912, −4.56310967323915203520061408250, −3.01431622136677926270850099423, −1.84450625891112027231292204991, 0.12628817676199198883819482195, 1.74857267653499751224449892942, 2.62959852555874738695377781622, 3.63789635508940738465154637537, 4.49061680501948392007643688702, 5.05879017885347499290228145477, 6.56992913462471261162359833899, 7.12981311247825660615007540672, 8.615082428656155849949911879872, 9.129886645239088572905366356500

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