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Properties

Label 1-145-145.37-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: $\chi_{145} (37, \cdot )$
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

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

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