↑

Properties

Label 1-119-119.13-r1-0-0
Degree $1$
Conductor $119$
Sign $-0.788 + 0.615i$
Analytic cond. $12.7883$
Root an. cond. $12.7883$
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  − 2-s − i·3-s + 4-s − i·5-s + i·6-s − 8-s − 9-s + i·10-s − i·11-s − i·12-s − 13-s − 15-s + 16-s + 18-s + 19-s − i·20-s + ⋯
L(s)  = 1  − 2-s − i·3-s + 4-s − i·5-s + i·6-s − 8-s − 9-s + i·10-s − i·11-s − i·12-s − 13-s − 15-s + 16-s + 18-s + 19-s − i·20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $-0.788 + 0.615i$
Analytic conductor: \(12.7883\)
Root analytic conductor: \(12.7883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 119,\ (1:\ ),\ -0.788 + 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1554999712 - 0.4518368174i\)
\(L(\frac12)\) \(\approx\) \(-0.1554999712 - 0.4518368174i\)
\(L(1)\) \(\approx\) \(0.4539895062 - 0.3544642952i\)
\(L(1)\) \(\approx\) \(0.4539895062 - 0.3544642952i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
17 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 - iT \)
19 \( 1 \)
23 \( 1 - T \)
29 \( 1 - T \)
31 \( 1 + iT \)
37 \( 1 - iT \)
41 \( 1 - iT \)
43 \( 1 - T \)
47 \( 1 \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 - iT \)
79 \( 1 \)
83 \( 1 + iT \)
89 \( 1 - iT \)
97 \( 1 + iT \)
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

−29.19429922096194439015139101662, −28.29658822080885250634100384886, −27.23577870720396442156449279465, −26.638732380569087590554438914299, −25.80458422713015182775086860627, −24.88783090966396096402792182385, −23.243143883059482442121433705752, −22.19789161233339904116157098793, −21.24571720642527163083354663787, −20.09392166383076148352682116221, −19.30939122599068426377812584296, −17.9031004325018123255645922781, −17.278704025654422354170254673843, −15.91025506896062865286363370778, −15.15879362855236650587071122325, −14.24761581104882481434150102413, −12.056098777009090978308056439857, −11.10516599267116146522075117958, −9.97121557408394636311508380876, −9.52364543811516615110626731275, −7.85795618022821410083317006889, −6.83978877535610179800140147366, −5.3050418431053638166819258547, −3.49117251284746848464952767623, −2.256241893414484609621788470523, 0.27242339148312720804332568658, 1.41746621829590072934334148496, 2.90676188138502215297918478979, 5.3060051230432239920409409431, 6.56395372244320502165746024061, 7.83445620791285445170904806461, 8.58864771748135478058126488006, 9.73658223446387255948633784700, 11.35628474101858382972412294939, 12.18042123951938075791156320422, 13.26443814480786611964631412244, 14.656175433988915348672906060, 16.29191185724555452708554417941, 16.89440721290865073476352301902, 17.98207033915929056157552725289, 18.94209750155176379662402074190, 19.86264785315769132496308553578, 20.60663050419687579626220505737, 22.04566136606730765946793612388, 23.7877339413209922792036482636, 24.43755308936762491209539650871, 25.01728157414247588729896414550, 26.288156098724667957196610388530, 27.28871186196980910773708219254, 28.4503693494321900283842120294

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