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Properties

Label 1-145-145.123-r1-0-0
Degree $1$
Conductor $145$
Sign $0.942 + 0.333i$
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 + (0.781 − 0.623i)7-s + (−0.974 − 0.222i)8-s + (0.222 − 0.974i)9-s + (−0.222 − 0.974i)11-s − i·12-s + (0.974 − 0.222i)13-s + (0.900 + 0.433i)14-s + (−0.222 − 0.974i)16-s − i·17-s + (0.974 − 0.222i)18-s + (−0.623 + 0.781i)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 + (0.781 − 0.623i)7-s + (−0.974 − 0.222i)8-s + (0.222 − 0.974i)9-s + (−0.222 − 0.974i)11-s − i·12-s + (0.974 − 0.222i)13-s + (0.900 + 0.433i)14-s + (−0.222 − 0.974i)16-s − i·17-s + (0.974 − 0.222i)18-s + (−0.623 + 0.781i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.942 + 0.333i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.942 + 0.333i$
Analytic conductor: \(15.5824\)
Root analytic conductor: \(15.5824\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (1:\ ),\ 0.942 + 0.333i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.436331765 + 0.2466619243i\)
\(L(\frac12)\) \(\approx\) \(1.436331765 + 0.2466619243i\)
\(L(1)\) \(\approx\) \(0.9571531428 + 0.4127851963i\)
\(L(1)\) \(\approx\) \(0.9571531428 + 0.4127851963i\)

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 \( 1 + (0.781 - 0.623i)T \)
11 \( 1 + (-0.222 - 0.974i)T \)
13 \( 1 + (0.974 - 0.222i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.623 + 0.781i)T \)
23 \( 1 + (0.433 - 0.900i)T \)
31 \( 1 + (-0.900 + 0.433i)T \)
37 \( 1 + (0.974 + 0.222i)T \)
41 \( 1 + T \)
43 \( 1 + (0.433 - 0.900i)T \)
47 \( 1 + (-0.974 + 0.222i)T \)
53 \( 1 + (-0.433 - 0.900i)T \)
59 \( 1 - T \)
61 \( 1 + (0.623 + 0.781i)T \)
67 \( 1 + (0.974 + 0.222i)T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (0.433 - 0.900i)T \)
79 \( 1 + (0.222 - 0.974i)T \)
83 \( 1 + (0.781 + 0.623i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
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

−27.98613606550198907173798548768, −27.74657177383050343801824809496, −25.922763366760875507346992796591, −24.65505312413982198617858359161, −23.621025521776764027039671814313, −23.15055934504712047915565755634, −21.84593435898391543469220622075, −21.24749685122118925400377912277, −19.961390524316673688766486142234, −18.88973131631103318255568011405, −18.04402880925672950875542053671, −17.33393472925094270730068943999, −15.57641481354792873359743091181, −14.55384868013145918367855603906, −13.1672652835205628994623355302, −12.56730456320636413475233925418, −11.351053412008296410005979548103, −10.865529855550630090841492764966, −9.327969382460257653946455976123, −7.96467932861972854415714227951, −6.340938206576841171210448616286, −5.31123264938614570690159694612, −4.23865977603775837555815256163, −2.28708691371923823913239326525, −1.32016383718193876466957218190, 0.60759396806264106672812551665, 3.4552146603460830305145420170, 4.5256559616151690885111776531, 5.55968032115690313474447577406, 6.55817736717718835890268810527, 7.93383165463786966323833345083, 9.0285124774165289564135412735, 10.60598319504024448399601130170, 11.462045899329491531023873212234, 12.81840176380246757126321504358, 13.99169126332675607375912567438, 14.90344728324111045563773858248, 16.18218818799329032817722733366, 16.591722422272089846715330157448, 17.77654718071183864834592159183, 18.531944644789016246364268889059, 20.6837558627486683698713737847, 21.18688268783908170945176882576, 22.3537675148237796247525397048, 23.23821419307785812460442002751, 23.87406022964119423758960754990, 24.94341012090952193296278181000, 26.20299748016640646656692033424, 27.12025287919228981548638058114, 27.56934278437468895413111030180

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