Properties

Label 1-37-37.3-r0-0-0
Degree $1$
Conductor $37$
Sign $-0.665 - 0.746i$
Analytic cond. $0.171827$
Root an. cond. $0.171827$
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  + (−0.173 − 0.984i)2-s + (0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)5-s − 6-s + (0.766 + 0.642i)7-s + (0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (0.173 + 0.984i)12-s + (0.939 − 0.342i)13-s + (0.5 − 0.866i)14-s + (−0.766 + 0.642i)15-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)5-s − 6-s + (0.766 + 0.642i)7-s + (0.5 + 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)11-s + (0.173 + 0.984i)12-s + (0.939 − 0.342i)13-s + (0.5 − 0.866i)14-s + (−0.766 + 0.642i)15-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(37\)
Sign: $-0.665 - 0.746i$
Analytic conductor: \(0.171827\)
Root analytic conductor: \(0.171827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 37,\ (0:\ ),\ -0.665 - 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2728970460 - 0.6091696721i\)
\(L(\frac12)\) \(\approx\) \(0.2728970460 - 0.6091696721i\)
\(L(1)\) \(\approx\) \(0.5730256230 - 0.6028616835i\)
\(L(1)\) \(\approx\) \(0.5730256230 - 0.6028616835i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
good2 \( 1 + (-0.173 - 0.984i)T \)
3 \( 1 + (0.173 - 0.984i)T \)
5 \( 1 + (-0.766 - 0.642i)T \)
7 \( 1 + (0.766 + 0.642i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.939 - 0.342i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 - T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.766 - 0.642i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.939 - 0.342i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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

−35.98481981469406901194170491145, −34.450878824919384579074961041155, −33.72370093807847105049095112730, −32.8104936441167454980384843732, −31.42460828509511208582358851621, −30.58374534777128762718162974694, −28.1568692134841556434653762626, −27.35234622302266576426852016860, −26.37811385004794953255824231267, −25.52412406137941537805807082375, −23.557185588307763619515639743372, −23.02220995021267017846663693034, −21.43418170723832982566322134725, −19.97289309798799357463800245099, −18.39327756383791419107886546660, −17.05280211805387660463422622819, −15.7187005940795798310575442379, −14.92992840018631673601381585473, −13.76162252981835000942418011962, −11.24575579035053140899260201945, −9.99471994821601930946177458734, −8.35020742300255258348190221368, −7.15317776955223512678826869163, −5.05529475560601806138394609841, −3.79770706941784267078472872641, 1.36111872145182545173201029804, 3.32290403676842290931058032293, 5.420362327351430206624935192594, 8.10272357707850378095747145648, 8.58139326078342803618355406973, 10.97700886914988791975009604334, 12.09429545145793591479221093531, 13.02738078356333233080063428918, 14.5248929595323202146958025779, 16.63398122301660007436649276387, 18.294280900585872254205705913206, 18.90854624061228455909847907802, 20.2857075846320586014774606501, 21.22532623007380492916163247380, 23.095673111181431982202952528895, 23.99340056835739397546274868979, 25.41089174075348964331022724642, 27.11102290279824701784820907542, 28.12653650299418180341509909885, 29.18124676862525714569439556429, 30.500940037908529412695328711730, 31.22779458708764400247871823403, 32.17589015975588054808924110867, 34.598115521742330046183640873689, 35.3246143635791424308720588999

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