↑

Properties

Label 2-832-104.101-c1-0-3
Degree $2$
Conductor $832$
Sign $0.494 - 0.869i$
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  + (−2.82 + 1.63i)3-s − 1.11·5-s + (−3.79 − 2.18i)7-s + (3.81 − 6.61i)9-s + (−1.09 − 1.89i)11-s + (1 + 3.46i)13-s + (3.15 − 1.81i)15-s + (0.942 − 1.63i)17-s + (2.05 − 3.56i)19-s + 14.2·21-s + (2.05 + 3.56i)23-s − 3.75·25-s + 15.1i·27-s + (−8.28 + 4.78i)29-s + 2i·31-s + ⋯
L(s)  = 1  + (−1.63 + 0.941i)3-s − 0.498·5-s + (−1.43 − 0.827i)7-s + (1.27 − 2.20i)9-s + (−0.329 − 0.570i)11-s + (0.277 + 0.960i)13-s + (0.813 − 0.469i)15-s + (0.228 − 0.395i)17-s + (0.472 − 0.818i)19-s + 3.11·21-s + (0.429 + 0.743i)23-s − 0.751·25-s + 2.91i·27-s + (−1.53 + 0.888i)29-s + 0.359i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $0.494 - 0.869i$
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: \((2,\ 832,\ (\ :1/2),\ 0.494 - 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.381200 + 0.221718i\)
\(L(\frac12)\) \(\approx\) \(0.381200 + 0.221718i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.82 - 1.63i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.11T + 5T^{2} \)
7 \( 1 + (3.79 + 2.18i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.09 + 1.89i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.942 + 1.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.05 + 3.56i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.05 - 3.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (8.28 - 4.78i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 + (1.20 + 2.08i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.17 + 3.56i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.892 - 0.515i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.11iT - 47T^{2} \)
53 \( 1 + 6.30iT - 53T^{2} \)
59 \( 1 + (-0.126 + 0.219i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.5 + 0.866i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.79 - 6.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-10.4 - 6.00i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.36iT - 73T^{2} \)
79 \( 1 - 5.90T + 79T^{2} \)
83 \( 1 - 14.7T + 83T^{2} \)
89 \( 1 + (-8.28 + 4.78i)T + (44.5 - 77.0i)T^{2} \)
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

−10.52373579668064855591024321359, −9.527381996864986956587130632782, −9.242531427299389420766610158718, −7.38539205015700027484341903611, −6.78978620394751888910468397233, −5.91151333684003944968030966180, −5.08930954486058471182581015713, −3.95616532641507565190306080839, −3.44583207735925992756979815829, −0.69647389123063145062076189388, 0.45814125644956027364944069401, 2.17508179707478980544665167586, 3.64348705212428179153206031237, 5.08506704271235924317696348899, 5.94320143288435730475186022703, 6.29681877004968036746832382115, 7.44891029544189423556271099367, 7.971420297164390696329267727239, 9.455178566554113151241622314938, 10.25519412001410546495349664659

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