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Properties

Label 1-145-145.84-r1-0-0
Degree $1$
Conductor $145$
Sign $0.857 - 0.514i$
Analytic cond. $15.5824$
Root an. cond. $15.5824$
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  + (0.433 + 0.900i)2-s + (0.781 − 0.623i)3-s + (−0.623 + 0.781i)4-s + (0.900 + 0.433i)6-s + (−0.623 − 0.781i)7-s + (−0.974 − 0.222i)8-s + (0.222 − 0.974i)9-s + (0.974 − 0.222i)11-s + i·12-s + (−0.222 − 0.974i)13-s + (0.433 − 0.900i)14-s + (−0.222 − 0.974i)16-s − i·17-s + (0.974 − 0.222i)18-s + (0.781 + 0.623i)19-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)2-s + (0.781 − 0.623i)3-s + (−0.623 + 0.781i)4-s + (0.900 + 0.433i)6-s + (−0.623 − 0.781i)7-s + (−0.974 − 0.222i)8-s + (0.222 − 0.974i)9-s + (0.974 − 0.222i)11-s + i·12-s + (−0.222 − 0.974i)13-s + (0.433 − 0.900i)14-s + (−0.222 − 0.974i)16-s − i·17-s + (0.974 − 0.222i)18-s + (0.781 + 0.623i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.857 - 0.514i$
Analytic conductor: \(15.5824\)
Root analytic conductor: \(15.5824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (1:\ ),\ 0.857 - 0.514i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.308662544 - 0.6395508403i\)
\(L(\frac12)\) \(\approx\) \(2.308662544 - 0.6395508403i\)
\(L(1)\) \(\approx\) \(1.532915203 + 0.05958206186i\)
\(L(1)\) \(\approx\) \(1.532915203 + 0.05958206186i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.433 + 0.900i)T \)
3 \( 1 + (0.781 - 0.623i)T \)
7 \( 1 + (-0.623 - 0.781i)T \)
11 \( 1 + (0.974 - 0.222i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.781 + 0.623i)T \)
23 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (-0.433 - 0.900i)T \)
37 \( 1 + (-0.974 - 0.222i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.433 + 0.900i)T \)
47 \( 1 + (0.974 - 0.222i)T \)
53 \( 1 + (0.900 - 0.433i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.781 + 0.623i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.433 - 0.900i)T \)
79 \( 1 + (-0.974 - 0.222i)T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (-0.433 - 0.900i)T \)
97 \( 1 + (0.781 + 0.623i)T \)
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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

−28.252453534159873365582205773983, −27.21107410706683633604442738276, −26.286240009188268168689481972582, −25.14706816541733541713380906312, −24.14707390925854144457440103390, −22.692219794764651069960378703197, −21.87986695188661651783546416889, −21.31610041586097438664639277557, −20.027786426535186414221857673283, −19.41593060014395156794152281014, −18.571872845722576194810106623563, −16.93592050516019844392455058235, −15.567579214780022233649307916907, −14.71679364185535726537936584618, −13.82124045122886456653688031935, −12.66379114327271944003991461242, −11.643401467716270427371800262346, −10.37936037723923160869018508517, −9.26897522962582279825041771155, −8.806715469540763474932988940942, −6.70454251910651342254803175217, −5.175130304078240554638390675534, −3.99794313283905971488734067016, −2.97395581878855134724016550417, −1.73360626991827547458521890467, 0.76221965105664081328848503788, 3.03635548552315046083341762691, 3.90671968658797435355183595360, 5.632498539655283567590535181622, 6.95348076145338541709340152991, 7.52138743540521720187229965344, 8.84197415753241847353531750964, 9.80454834408405860429381954382, 11.83471850155406449529403769102, 12.93770026342054905927104686223, 13.726438541887909364139297738831, 14.52559775563642419090922574348, 15.62139588575917029662470753487, 16.755658986984237026216658143425, 17.72160819295942159632637464498, 18.850757632501686897411242772665, 19.96996891830226293947665851080, 20.85710982204754086556324796629, 22.426556916304675769569893637253, 22.99252729545546555269706716013, 24.22199638024947903683918432863, 24.96465370173894673229884420236, 25.65211969469222518527973828457, 26.74173038907850771185753460816, 27.37302509378858530559120920833

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