↑

Properties

Label 2-832-104.101-c1-0-18
Degree $2$
Conductor $832$
Sign $0.869 + 0.494i$
Analytic cond. $6.64355$
Root an. cond. $2.57750$
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  + (1.16 − 0.669i)3-s + 3.88·5-s + (−2.20 − 1.27i)7-s + (−0.602 + 1.04i)9-s + (−0.571 − 0.990i)11-s + (1 + 3.46i)13-s + (4.50 − 2.60i)15-s + (3.44 − 5.96i)17-s + (3.93 − 6.81i)19-s − 3.40·21-s + (3.93 + 6.81i)23-s + 10.0·25-s + 5.63i·27-s + (−2.51 + 1.45i)29-s − 2i·31-s + ⋯
L(s)  = 1  + (0.669 − 0.386i)3-s + 1.73·5-s + (−0.832 − 0.480i)7-s + (−0.200 + 0.347i)9-s + (−0.172 − 0.298i)11-s + (0.277 + 0.960i)13-s + (1.16 − 0.671i)15-s + (0.834 − 1.44i)17-s + (0.902 − 1.56i)19-s − 0.744·21-s + (0.820 + 1.42i)23-s + 2.01·25-s + 1.08i·27-s + (−0.467 + 0.270i)29-s − 0.359i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $0.869 + 0.494i$
Analytic conductor: \(6.64355\)
Root analytic conductor: \(2.57750\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :1/2),\ 0.869 + 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.33496 - 0.617636i\)
\(L(\frac12)\) \(\approx\) \(2.33496 - 0.617636i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-1 - 3.46i)T \)
good3 \( 1 + (-1.16 + 0.669i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 - 3.88T + 5T^{2} \)
7 \( 1 + (2.20 + 1.27i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.571 + 0.990i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.44 + 5.96i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.93 + 6.81i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.93 - 6.81i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.51 - 1.45i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 2iT - 31T^{2} \)
37 \( 1 + (1.78 + 3.08i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.32 - 0.765i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.88 + 4.55i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 2.11iT - 47T^{2} \)
53 \( 1 - 9.01iT - 53T^{2} \)
59 \( 1 + (2.79 - 4.83i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.5 + 0.866i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.20 - 3.81i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.24 - 1.87i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.3iT - 73T^{2} \)
79 \( 1 + 7.90T + 79T^{2} \)
83 \( 1 + 8.10T + 83T^{2} \)
89 \( 1 + (-2.51 + 1.45i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.5 - 0.866i)T + (48.5 + 84.0i)T^{2} \)
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

−9.755159962108812489103450281697, −9.423442935699565624874176571896, −8.754802161430476856232624512143, −7.21630139756258986465983299841, −6.98855079199724988041186609847, −5.66293996122109900123101248699, −5.08229428260084848720907617240, −3.26921829795448337146472742727, −2.57110836353210612464246590619, −1.32546201315778917061428781216, 1.58485609718571830350414076191, 2.86837494947179096262979113107, 3.50605576327557643668248723002, 5.18163529754914898832567140484, 5.99858244959645947669211531268, 6.44199924181915001488702170303, 8.038014489630476176390560621871, 8.715766237538948984127176885873, 9.690287803272624712761754973841, 9.971537513020996681473802269725

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