↑

Properties

Label 1-163-163.15-r0-0-0
Degree $1$
Conductor $163$
Sign $-0.995 + 0.0967i$
Analytic cond. $0.756968$
Root an. cond. $0.756968$
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  + (−0.211 − 0.977i)2-s + (−0.431 + 0.902i)3-s + (−0.910 + 0.413i)4-s + (−0.0581 − 0.998i)5-s + (0.973 + 0.230i)6-s + (−0.135 + 0.990i)7-s + (0.597 + 0.802i)8-s + (−0.627 − 0.778i)9-s + (−0.963 + 0.268i)10-s + (−0.565 − 0.824i)11-s + (0.0193 − 0.999i)12-s + (−0.993 − 0.116i)13-s + (0.996 − 0.0774i)14-s + (0.925 + 0.378i)15-s + (0.657 − 0.753i)16-s + (0.396 − 0.918i)17-s + ⋯
L(s)  = 1  + (−0.211 − 0.977i)2-s + (−0.431 + 0.902i)3-s + (−0.910 + 0.413i)4-s + (−0.0581 − 0.998i)5-s + (0.973 + 0.230i)6-s + (−0.135 + 0.990i)7-s + (0.597 + 0.802i)8-s + (−0.627 − 0.778i)9-s + (−0.963 + 0.268i)10-s + (−0.565 − 0.824i)11-s + (0.0193 − 0.999i)12-s + (−0.993 − 0.116i)13-s + (0.996 − 0.0774i)14-s + (0.925 + 0.378i)15-s + (0.657 − 0.753i)16-s + (0.396 − 0.918i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(163\)
Sign: $-0.995 + 0.0967i$
Analytic conductor: \(0.756968\)
Root analytic conductor: \(0.756968\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{163} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 163,\ (0:\ ),\ -0.995 + 0.0967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.009559006235 - 0.1970541124i\)
\(L(\frac12)\) \(\approx\) \(0.009559006235 - 0.1970541124i\)
\(L(1)\) \(\approx\) \(0.4729680619 - 0.1959234868i\)
\(L(1)\) \(\approx\) \(0.4729680619 - 0.1959234868i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad163 \( 1 \)
good2 \( 1 + (-0.211 - 0.977i)T \)
3 \( 1 + (-0.431 + 0.902i)T \)
5 \( 1 + (-0.0581 - 0.998i)T \)
7 \( 1 + (-0.135 + 0.990i)T \)
11 \( 1 + (-0.565 - 0.824i)T \)
13 \( 1 + (-0.993 - 0.116i)T \)
17 \( 1 + (0.396 - 0.918i)T \)
19 \( 1 + (-0.999 - 0.0387i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.740 - 0.672i)T \)
31 \( 1 + (-0.835 + 0.549i)T \)
37 \( 1 + (-0.686 - 0.727i)T \)
41 \( 1 + (-0.910 - 0.413i)T \)
43 \( 1 + (0.249 + 0.968i)T \)
47 \( 1 + (0.952 + 0.305i)T \)
53 \( 1 + (0.766 + 0.642i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.597 - 0.802i)T \)
67 \( 1 + (0.813 - 0.581i)T \)
71 \( 1 + (0.713 - 0.700i)T \)
73 \( 1 + (0.813 + 0.581i)T \)
79 \( 1 + (0.987 + 0.154i)T \)
83 \( 1 + (-0.790 - 0.612i)T \)
89 \( 1 + (-0.565 + 0.824i)T \)
97 \( 1 + (-0.360 - 0.932i)T \)
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

−28.089361362819686835548360529990, −27.09385511460866101624850072489, −25.89671691823086543775982989912, −25.67465689675849228028367057716, −24.06237784974922740416141712058, −23.63810921524559705706967541111, −22.730027953590236560905987609328, −21.978859516475263021664067580618, −19.99024456881047605960563801912, −19.06122714718024723879555015202, −18.25442498427316308636733383051, −17.29882684260584254590067031822, −16.704284584282601724282509734578, −15.13484593583778255502420049848, −14.37911571980194473731099219471, −13.34720018939171388585017122992, −12.35633226918089158436302274448, −10.66967543669672657694978270782, −10.04437147779619727513938078291, −8.149097582502755392687470131245, −7.25863241702297937372544872308, −6.72953739247123583792626115360, −5.48545456882722443428828709468, −4.0152957201300781120093814923, −2.0041738734647651631040566976, 0.17569631184797394753752991580, 2.317112633902251743132067758737, 3.67772011084996094404951628289, 4.98167315711602971306532349798, 5.61960073434098495760794008177, 8.08214112868098353589049289406, 9.08151719573995058810915866114, 9.75740129946545155353413156905, 11.00042020514561224933672879182, 12.01419238855072630572755848500, 12.65189549922865197383489374331, 14.07239860356591047626702633295, 15.4824330169086817977720690964, 16.45618390102795678024066978866, 17.314076370606872978427547787369, 18.45636691403405445473344517757, 19.56220790793336051012943473716, 20.56750215933777805168337878176, 21.41991718544391704777598117061, 21.931452821525194593927431080546, 23.0719466738881778431207405515, 24.19790925915494675265802437531, 25.52057438334811095434049182574, 26.686409397478142416139135966709, 27.5951838535031393612381462253

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