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Properties

Label 1-119-119.90-r0-0-0
Degree $1$
Conductor $119$
Sign $0.986 - 0.163i$
Analytic cond. $0.552633$
Root an. cond. $0.552633$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.382 − 0.923i)3-s − i·4-s + (0.923 + 0.382i)5-s + (0.382 + 0.923i)6-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.923 + 0.382i)10-s + (0.382 + 0.923i)11-s + (−0.923 − 0.382i)12-s − i·13-s + (0.707 − 0.707i)15-s − 16-s + 18-s + (0.707 − 0.707i)19-s + (0.382 − 0.923i)20-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (0.382 − 0.923i)3-s − i·4-s + (0.923 + 0.382i)5-s + (0.382 + 0.923i)6-s + (0.707 + 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.923 + 0.382i)10-s + (0.382 + 0.923i)11-s + (−0.923 − 0.382i)12-s − i·13-s + (0.707 − 0.707i)15-s − 16-s + 18-s + (0.707 − 0.707i)19-s + (0.382 − 0.923i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(119\)    =    \(7 \cdot 17\)
Sign: $0.986 - 0.163i$
Analytic conductor: \(0.552633\)
Root analytic conductor: \(0.552633\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{119} (90, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 119,\ (0:\ ),\ 0.986 - 0.163i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9593000856 - 0.07909924429i\)
\(L(\frac12)\) \(\approx\) \(0.9593000856 - 0.07909924429i\)
\(L(1)\) \(\approx\) \(0.9557105945 + 0.003732628473i\)
\(L(1)\) \(\approx\) \(0.9557105945 + 0.003732628473i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.382 - 0.923i)T \)
5 \( 1 + (0.923 + 0.382i)T \)
11 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 - iT \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (-0.382 - 0.923i)T \)
29 \( 1 + (0.923 + 0.382i)T \)
31 \( 1 + (-0.382 + 0.923i)T \)
37 \( 1 + (-0.382 + 0.923i)T \)
41 \( 1 + (0.923 - 0.382i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.923 + 0.382i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.382 + 0.923i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.923 - 0.382i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−28.99310987457623086959704389178, −28.16128504847897368744345261820, −27.137396976150572459317443321398, −26.33865360690165440795303019682, −25.46956252152850946683782783716, −24.46749991298573353706879311618, −22.52386337984658414924190898305, −21.48720977216615198753535780399, −21.158888607104501303541143402768, −19.99662960914826305459936757497, −19.059275405819543377820798508310, −17.75551149126064899858106922769, −16.66923479099192151582784553132, −16.10125684939619325431423326931, −14.23598417639681201873003597251, −13.4793820382163669262619548320, −11.88860098979727665046755158793, −10.83390620159696930372480754740, −9.6304606696070076986243692532, −9.146306405936592761384951888994, −7.943397066862626819745161158478, −5.99460944303988107565215109505, −4.40001897843951705211950598490, −3.13726124075247026000691575372, −1.69593358449688567603491887831, 1.36721400122748507345868577069, 2.68161852031219285634003598489, 5.222347716115456942324269930211, 6.472470441031297672457858814805, 7.2195000978460313219215003148, 8.48477723024456482139890025732, 9.57017291692260215909553256753, 10.6350656066202540953894022623, 12.33322246684679799008535781362, 13.62390997083856891955717284415, 14.440131745084217548753214826, 15.42754234898759480232464910661, 17.06216747055900920097337581874, 17.8761611946330583068427966957, 18.37867031869913418086110937029, 19.73882779515577572690252714770, 20.44916259114906085589883343247, 22.287166213882401324200352182, 23.23566625948499955943118830938, 24.46302268890246165336519946127, 25.18592886058967320013769253437, 25.79182073889937300062440581097, 26.78529496683660446975407416423, 28.16827055162801820063667396084, 29.02039318934919241430938959683

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