Properties

Label 2-1617-77.76-c1-0-11
Degree $2$
Conductor $1617$
Sign $0.0396 + 0.999i$
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  + 2.34i·2-s i·3-s − 3.51·4-s + 3.84i·5-s + 2.34·6-s − 3.56i·8-s − 9-s − 9.02·10-s + (0.648 + 3.25i)11-s + 3.51i·12-s − 2.56·13-s + 3.84·15-s + 1.33·16-s − 0.436·17-s − 2.34i·18-s + 5.74·19-s + ⋯
L(s)  = 1  + 1.66i·2-s − 0.577i·3-s − 1.75·4-s + 1.71i·5-s + 0.958·6-s − 1.25i·8-s − 0.333·9-s − 2.85·10-s + (0.195 + 0.980i)11-s + 1.01i·12-s − 0.710·13-s + 0.992·15-s + 0.333·16-s − 0.105·17-s − 0.553i·18-s + 1.31·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0396 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0396 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.0396 + 0.999i$
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: \((2,\ 1617,\ (\ :1/2),\ 0.0396 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8326659688\)
\(L(\frac12)\) \(\approx\) \(0.8326659688\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.648 - 3.25i)T \)
good2 \( 1 - 2.34iT - 2T^{2} \)
5 \( 1 - 3.84iT - 5T^{2} \)
13 \( 1 + 2.56T + 13T^{2} \)
17 \( 1 + 0.436T + 17T^{2} \)
19 \( 1 - 5.74T + 19T^{2} \)
23 \( 1 + 5.71T + 23T^{2} \)
29 \( 1 + 1.22iT - 29T^{2} \)
31 \( 1 - 5.39iT - 31T^{2} \)
37 \( 1 + 9.95T + 37T^{2} \)
41 \( 1 - 12.3T + 41T^{2} \)
43 \( 1 - 8.05iT - 43T^{2} \)
47 \( 1 + 5.43iT - 47T^{2} \)
53 \( 1 - 6.51T + 53T^{2} \)
59 \( 1 + 3.88iT - 59T^{2} \)
61 \( 1 + 4.47T + 61T^{2} \)
67 \( 1 + 4.08T + 67T^{2} \)
71 \( 1 + 5.05T + 71T^{2} \)
73 \( 1 + 3.52T + 73T^{2} \)
79 \( 1 + 16.6iT - 79T^{2} \)
83 \( 1 + 4.69T + 83T^{2} \)
89 \( 1 + 2.88iT - 89T^{2} \)
97 \( 1 + 11.2iT - 97T^{2} \)
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

−9.841297539147775350930245607294, −9.010958185968493427752130616786, −7.81143870986897051613756897515, −7.42920035675969026206575341838, −6.93072379358039479598455712706, −6.25637004867581760313456377213, −5.49311720315632277585012452108, −4.42973032134671793811731108066, −3.21337499660328804243582383065, −2.09551767577177379746274894310, 0.33301295550778353975874003233, 1.34143771013260094433264890804, 2.55124581822358110762171597399, 3.71142546534119488804110129032, 4.29176646530014638903284783686, 5.18378156205515207458847033167, 5.79713482057810206081654394182, 7.57176599577507050278731829738, 8.546688903295579002925563648871, 9.076105040402579055205252208464

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