↑

Properties

Label 2-29e2-29.20-c1-0-49
Degree $2$
Conductor $841$
Sign $-0.349 + 0.936i$
Analytic cond. $6.71541$
Root an. cond. $2.59141$
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  + (1.62 − 0.781i)2-s + (0.277 + 0.347i)3-s + (0.777 − 0.974i)4-s + (−0.321 + 0.154i)5-s + (0.722 + 0.347i)6-s + (−2.52 − 3.16i)7-s + (−0.301 + 1.32i)8-s + (0.623 − 2.73i)9-s + (−0.400 + 0.502i)10-s + (−0.647 − 2.83i)11-s + 0.554·12-s + (−1.15 − 5.05i)13-s + (−6.57 − 3.16i)14-s + (−0.143 − 0.0689i)15-s + (1.09 + 4.81i)16-s + 1.10·17-s + ⋯
L(s)  = 1  + (1.14 − 0.552i)2-s + (0.160 + 0.200i)3-s + (0.388 − 0.487i)4-s + (−0.143 + 0.0692i)5-s + (0.294 + 0.142i)6-s + (−0.954 − 1.19i)7-s + (−0.106 + 0.467i)8-s + (0.207 − 0.910i)9-s + (−0.126 + 0.158i)10-s + (−0.195 − 0.855i)11-s + 0.160·12-s + (−0.320 − 1.40i)13-s + (−1.75 − 0.846i)14-s + (−0.0369 − 0.0177i)15-s + (0.274 + 1.20i)16-s + 0.269·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.349 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.349 + 0.936i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ -0.349 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29346 - 1.86406i\)
\(L(\frac12)\) \(\approx\) \(1.29346 - 1.86406i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
good2 \( 1 + (-1.62 + 0.781i)T + (1.24 - 1.56i)T^{2} \)
3 \( 1 + (-0.277 - 0.347i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (0.321 - 0.154i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (2.52 + 3.16i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (0.647 + 2.83i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (1.15 + 5.05i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 - 1.10T + 17T^{2} \)
19 \( 1 + (-1.27 + 1.60i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (-3.72 - 1.79i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (-5.72 + 2.75i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + (-0.647 + 2.83i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 - 0.396T + 41T^{2} \)
43 \( 1 + (5.17 + 2.49i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-1.73 - 7.60i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-3.92 + 1.89i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + 9.10T + 59T^{2} \)
61 \( 1 + (3.77 + 4.72i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (0.0833 - 0.364i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (-2.53 - 11.1i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-8.06 - 3.88i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + (-0.132 + 0.579i)T + (-71.1 - 34.2i)T^{2} \)
83 \( 1 + (5.88 - 7.37i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (1.28 - 0.617i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (-9.82 + 12.3i)T + (-21.5 - 94.5i)T^{2} \)
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

−10.12826146259148006563669422924, −9.348631832896686485279798042891, −8.170982185212781799739372852678, −7.26074049231846096177792707687, −6.20987464199083468099110636309, −5.37017694936412850852892198725, −4.22021467461474394023228591255, −3.34846533694235332624529778506, −3.02384497990792850262154565277, −0.75164313535719559817012404766, 2.10180907355615245181884661058, 3.14349323134961595398402788973, 4.46612930349359788768235640390, 5.03247739848568464884486130889, 6.11500886247677900661314648452, 6.76340028160534542975426793530, 7.60326368353054029403364632651, 8.753641573461183523802901312245, 9.615090364716420121743539162608, 10.27518933326596812524864462103

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