↑

Properties

Label 1-185-185.79-r1-0-0
Degree $1$
Conductor $185$
Sign $-0.845 + 0.534i$
Analytic cond. $19.8810$
Root an. cond. $19.8810$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.642 + 0.766i)2-s + (0.766 + 0.642i)3-s + (−0.173 + 0.984i)4-s + i·6-s + (0.939 − 0.342i)7-s + (−0.866 + 0.5i)8-s + (0.173 + 0.984i)9-s + (0.5 + 0.866i)11-s + (−0.766 + 0.642i)12-s + (−0.984 − 0.173i)13-s + (0.866 + 0.5i)14-s + (−0.939 − 0.342i)16-s + (0.984 − 0.173i)17-s + (−0.642 + 0.766i)18-s + (−0.642 + 0.766i)19-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (0.766 + 0.642i)3-s + (−0.173 + 0.984i)4-s + i·6-s + (0.939 − 0.342i)7-s + (−0.866 + 0.5i)8-s + (0.173 + 0.984i)9-s + (0.5 + 0.866i)11-s + (−0.766 + 0.642i)12-s + (−0.984 − 0.173i)13-s + (0.866 + 0.5i)14-s + (−0.939 − 0.342i)16-s + (0.984 − 0.173i)17-s + (−0.642 + 0.766i)18-s + (−0.642 + 0.766i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.845 + 0.534i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ -0.845 + 0.534i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9588991992 + 3.310028790i\)
\(L(\frac12)\) \(\approx\) \(0.9588991992 + 3.310028790i\)
\(L(1)\) \(\approx\) \(1.374636916 + 1.428880632i\)
\(L(1)\) \(\approx\) \(1.374636916 + 1.428880632i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.642 + 0.766i)T \)
3 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (0.939 - 0.342i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.984 - 0.173i)T \)
17 \( 1 + (0.984 - 0.173i)T \)
19 \( 1 + (-0.642 + 0.766i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.939 + 0.342i)T \)
59 \( 1 + (0.342 - 0.939i)T \)
61 \( 1 + (-0.984 - 0.173i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.342 - 0.939i)T \)
83 \( 1 + (-0.173 - 0.984i)T \)
89 \( 1 + (-0.342 + 0.939i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−26.81513151763712429044653334395, −25.35382213743519581796363990283, −24.30598163682895748413152520554, −24.03581086023180559499931299820, −22.70935061658024904185563801177, −21.42558218171263892597064065712, −21.032407200102892166162740819182, −19.74000318085161730189003110962, −19.1482285026532054484727561193, −18.26710915166265941039778849099, −17.04189372748116096909349131555, −15.19497425234963117966336036446, −14.54275242892681940826114117007, −13.812247813871219136623512920993, −12.64805883601121385444333926974, −11.86916047596573249052899613787, −10.823873617363686015943725408042, −9.37433782060385014394395585022, −8.505403944816888187473296550890, −7.12726806807564125503709448228, −5.79409372918168650242731028387, −4.51613990404691029499381527783, −3.176406644891687186448783386008, −2.13215471456899006329609957760, −0.927686472076444308434567222944, 2.07278311061844451922606004687, 3.56050207124454190200832900381, 4.54415739268640805906790019196, 5.40731876698089029838703586753, 7.23867963777088278714430674406, 7.85042974441208396762174131210, 9.0513139597618357152679520197, 10.16712855481790574988513200774, 11.63035979552633374037281781488, 12.77487618681243615582457067445, 13.97287214958718987816551393414, 14.82543415410285018843407038979, 15.1165149214506398163027567738, 16.72462969731631915619960666134, 17.14788884053658835767556492175, 18.6039200680612003978438550391, 20.05167648549046607032061807879, 20.8193856096459276461826435705, 21.614580398851941465433138933546, 22.61222147604976721474169645537, 23.582022078951605583236977516972, 24.75367832852079353446313124607, 25.28425965591097998735362378495, 26.28055300978230065875329441246, 27.287762732611871784532929836218

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