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Properties

Label 1-145-145.98-r0-0-0
Degree $1$
Conductor $145$
Sign $0.352 + 0.935i$
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  + (−0.623 − 0.781i)2-s + (−0.222 − 0.974i)3-s + (−0.222 + 0.974i)4-s + (−0.623 + 0.781i)6-s + (−0.974 + 0.222i)7-s + (0.900 − 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.433 + 0.900i)11-s + 12-s + (−0.433 + 0.900i)13-s + (0.781 + 0.623i)14-s + (−0.900 − 0.433i)16-s − 17-s + (0.900 + 0.433i)18-s + (−0.974 − 0.222i)19-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)2-s + (−0.222 − 0.974i)3-s + (−0.222 + 0.974i)4-s + (−0.623 + 0.781i)6-s + (−0.974 + 0.222i)7-s + (0.900 − 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.433 + 0.900i)11-s + 12-s + (−0.433 + 0.900i)13-s + (0.781 + 0.623i)14-s + (−0.900 − 0.433i)16-s − 17-s + (0.900 + 0.433i)18-s + (−0.974 − 0.222i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.352 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(145\)    =    \(5 \cdot 29\)
Sign: $0.352 + 0.935i$
Analytic conductor: \(0.673377\)
Root analytic conductor: \(0.673377\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{145} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 145,\ (0:\ ),\ 0.352 + 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1589454782 + 0.1099218802i\)
\(L(\frac12)\) \(\approx\) \(0.1589454782 + 0.1099218802i\)
\(L(1)\) \(\approx\) \(0.4324809639 - 0.1604872038i\)
\(L(1)\) \(\approx\) \(0.4324809639 - 0.1604872038i\)

Euler product

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

−27.8016366174387232700281597069, −26.88290959825005549312271935941, −26.3040956933431301163275785954, −25.347471002314940755888935892557, −24.23713117543043147116126174063, −22.981304297029696230275156335634, −22.44523741513279094388748622667, −21.112650584907911192406942577866, −19.863699951087716603426672570641, −19.07855437716988118496480720132, −17.66765419135040171744681519840, −16.84235121019466835346926748450, −15.926737077398731645904205492198, −15.30107214958946505713958545508, −14.094282863163264534696501754257, −12.818857074289305236707430940628, −10.90380169035086133050843111217, −10.347310047818689159251842006358, −9.19514304108715113732979309118, −8.29764767196920007944914862658, −6.69690999195764074238559838168, −5.75062698486340898021406033086, −4.54403696510168022659960631867, −2.963954449447317989733951602101, −0.19969622341558657549133531057, 1.84231011182217444801707310933, 2.82348555370001283069310945859, 4.58579598509946250967567663104, 6.50820677609855050656331686432, 7.32061565557999557704962424841, 8.68032259048481324527618109278, 9.67562763752738404313170805760, 10.96100088923093423921061607289, 12.04859279718363367025964906549, 12.844739434358901662509501037272, 13.61456209647808470108049117818, 15.409504254926783831837476702871, 16.82175656986289179425749127002, 17.52569008235666729981026359519, 18.65148213467837883701215229229, 19.31523411870386245190792482770, 20.10244037715561912505396105396, 21.43668819803530204682146269919, 22.49819357134417465636860126072, 23.31827377601770501225613014146, 24.67147603697175241815776818247, 25.725060619318741640636318786, 26.29057510303633547631273784073, 27.786800009384928931833705397178, 28.72925212052469117069367118145

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