Properties

Label 1-145-145.32-r0-0-0
Degree $1$
Conductor $145$
Sign $0.638 + 0.769i$
Analytic cond. $0.673377$
Root an. cond. $0.673377$
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  + (−0.900 + 0.433i)2-s + (−0.623 − 0.781i)3-s + (0.623 − 0.781i)4-s + (0.900 + 0.433i)6-s + (−0.781 + 0.623i)7-s + (−0.222 + 0.974i)8-s + (−0.222 + 0.974i)9-s + (−0.974 + 0.222i)11-s − 12-s + (0.974 − 0.222i)13-s + (0.433 − 0.900i)14-s + (−0.222 − 0.974i)16-s + 17-s + (−0.222 − 0.974i)18-s + (0.781 + 0.623i)19-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (−0.623 − 0.781i)3-s + (0.623 − 0.781i)4-s + (0.900 + 0.433i)6-s + (−0.781 + 0.623i)7-s + (−0.222 + 0.974i)8-s + (−0.222 + 0.974i)9-s + (−0.974 + 0.222i)11-s − 12-s + (0.974 − 0.222i)13-s + (0.433 − 0.900i)14-s + (−0.222 − 0.974i)16-s + 17-s + (−0.222 − 0.974i)18-s + (0.781 + 0.623i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.638 + 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.638 + 0.769i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ 0.638 + 0.769i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4390529236 + 0.2061234152i\)
\(L(\frac12)\) \(\approx\) \(0.4390529236 + 0.2061234152i\)
\(L(1)\) \(\approx\) \(0.5415971205 + 0.07537696763i\)
\(L(1)\) \(\approx\) \(0.5415971205 + 0.07537696763i\)

Euler product

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

−28.214447347406379067676719426297, −27.17570663026771232882531580417, −26.153560580194742503487720459831, −25.873179264237395654564789386670, −24.08571942408525905843350768015, −23.00037061408331617283461313665, −22.026925627774719924846773828814, −20.87569598518024450878405833485, −20.37641042102158247481689518347, −18.95066563418460693301022282865, −18.113984438582011172955145038415, −16.91555060855207547585544054887, −16.21959709463415528225833257616, −15.49486177902767734579620127464, −13.6137665950484551121398679955, −12.39108672871553873113609387440, −11.246816850199728966104065678465, −10.359240863126278227696004570063, −9.66318503172533508863029073584, −8.384544129189798844049470925927, −6.993200395438770634235673432220, −5.774647997183560019503353697514, −4.02363722254709135458578729605, −2.96020064475655157917430111851, −0.69722122096291109218809976184, 1.29147269885104512653275952122, 2.84754661959424162488215518693, 5.48231644201104952280405108353, 6.04392902704947375461929605269, 7.37956242006489890546788901769, 8.20939895890610382518720824785, 9.63344588854584405914430879509, 10.663520369799748892372631218729, 11.83962878287618072269230171339, 12.87398798913468279867431098355, 14.149460870166852728758472614143, 15.79956690064335894661593966243, 16.18218306472491998931652677851, 17.541304798978014000674677160363, 18.38933438848308229338766145222, 18.929961602346640273057880196718, 20.06147397361593598579566721491, 21.383312039830090944845932821727, 22.99759076885856307364988791903, 23.3939568171089613391574151481, 24.69646302465557954161422835596, 25.4141799655280426458541995801, 26.18983576328997064841728886292, 27.64755669897814302947960908909, 28.39153489311874202977292925846

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