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Properties

Label 1-145-145.73-r0-0-0
Degree $1$
Conductor $145$
Sign $0.999 + 0.0104i$
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  + (−0.222 − 0.974i)2-s + (0.900 + 0.433i)3-s + (−0.900 + 0.433i)4-s + (0.222 − 0.974i)6-s + (−0.433 + 0.900i)7-s + (0.623 + 0.781i)8-s + (0.623 + 0.781i)9-s + (0.781 + 0.623i)11-s − 12-s + (−0.781 − 0.623i)13-s + (0.974 + 0.222i)14-s + (0.623 − 0.781i)16-s + 17-s + (0.623 − 0.781i)18-s + (0.433 + 0.900i)19-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (0.900 + 0.433i)3-s + (−0.900 + 0.433i)4-s + (0.222 − 0.974i)6-s + (−0.433 + 0.900i)7-s + (0.623 + 0.781i)8-s + (0.623 + 0.781i)9-s + (0.781 + 0.623i)11-s − 12-s + (−0.781 − 0.623i)13-s + (0.974 + 0.222i)14-s + (0.623 − 0.781i)16-s + 17-s + (0.623 − 0.781i)18-s + (0.433 + 0.900i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.999 + 0.0104i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ 0.999 + 0.0104i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.173774499 + 0.006143413569i\)
\(L(\frac12)\) \(\approx\) \(1.173774499 + 0.006143413569i\)
\(L(1)\) \(\approx\) \(1.118214321 - 0.1294620355i\)
\(L(1)\) \(\approx\) \(1.118214321 - 0.1294620355i\)

Euler product

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

−27.9307440567690908763976702688, −26.7276290086205305476016625436, −26.32995427660499871415831933906, −25.31896569508314505637986260274, −24.35212608919669432435865356519, −23.746406507030613140757338953486, −22.54935754748434149471873304069, −21.36322812493989624251461860464, −19.73514790050953368002339493074, −19.40093970099664187481899009616, −18.18252062011423706316862778357, −17.0333888162593545819565275961, −16.19641275158917264856916248876, −14.91057653350830636780640086339, −13.9967871956994510595164635186, −13.465766554539832498850553789123, −12.013044787654771112892003826727, −10.05775750660714709347631688709, −9.30311694595888913489036854753, −8.09954217451959707857182497129, −7.15928512965866031999055413843, −6.305993945359895766141834092946, −4.499377673984529933175833310102, −3.30695488623981150232625368494, −1.1732506477054781640677806882, 1.87596768728301516838851440956, 2.98297942612270914446881942016, 4.05089618081459686162024239901, 5.45182936631635922090905976654, 7.56977341999181714466104150575, 8.63017330608357021056887738024, 9.69310497542098869002448796731, 10.20223200915610282751708596732, 11.99834455129620277971115412901, 12.56152650257527026709098723028, 14.00558051685693612195764988986, 14.79660978145282831856067525508, 16.07382157388214627044854757787, 17.3733692672179755798558328066, 18.64154918464044999599321731833, 19.37957947791388265477705292443, 20.25420731719220881341817607606, 21.12009017890285658748872272090, 22.15149046802300050028380161013, 22.73694375140004703543737684587, 24.65943050038710042630946843603, 25.470498345227693833487506526273, 26.3859636298775081380858807027, 27.55708991035817332529039093820, 27.94355595708179460574404972677

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