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Properties

Label 1-240-240.149-r1-0-0
Degree $1$
Conductor $240$
Sign $0.923 - 0.382i$
Analytic cond. $25.7915$
Root an. cond. $25.7915$
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  + 7-s − i·11-s + i·13-s + 17-s − i·19-s − 23-s + i·29-s + 31-s − i·37-s + 41-s − i·43-s + 47-s + 49-s + i·53-s − i·59-s + ⋯
L(s)  = 1  + 7-s − i·11-s + i·13-s + 17-s − i·19-s − 23-s + i·29-s + 31-s − i·37-s + 41-s − i·43-s + 47-s + 49-s + i·53-s − i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(25.7915\)
Root analytic conductor: \(25.7915\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 240,\ (1:\ ),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.146357629 - 0.4269370774i\)
\(L(\frac12)\) \(\approx\) \(2.146357629 - 0.4269370774i\)
\(L(1)\) \(\approx\) \(1.279323202 - 0.09091870067i\)
\(L(1)\) \(\approx\) \(1.279323202 - 0.09091870067i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 + T \)
23 \( 1 \)
29 \( 1 \)
31 \( 1 \)
37 \( 1 - iT \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 + T \)
67 \( 1 \)
71 \( 1 - iT \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 \)
89 \( 1 - T \)
97 \( 1 \)
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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

−25.90371201405434642710887103764, −25.0844728381234795818615395684, −24.26267166760179045253361914737, −23.1481673189107644995109667752, −22.52336229409206171130631185482, −21.12169492183392830705876617405, −20.64994615694929909418919854541, −19.59548255064099420947054908225, −18.35552614386931539457382160332, −17.686273714267341723725249420627, −16.76051551313030989037590362337, −15.46599704806247463179071810946, −14.71787631255747776317827255120, −13.789154211821039035903283842840, −12.46892239634362715863328125311, −11.78469901406783316647916490794, −10.45536201253392721722872556759, −9.73959358382744246107579561300, −8.114104268125775780993047062657, −7.700990048890870998839710282077, −6.10472745145157589365582024133, −5.05014312191771242704643256839, −3.942241158985500683191562847224, −2.405462218965854102224080933232, −1.1175233438145470490536700760, 0.88660265132321572416727806684, 2.26455066210037854678303013703, 3.744384072034938805327123857842, 4.91865821276639257433735218190, 6.00855184349712910733598996818, 7.308700592461037135144111062489, 8.345216703447689818223413364097, 9.2626495609544290336554100993, 10.64815975480429094977978286835, 11.45278241327108232879908354104, 12.37176326832992043865724501860, 13.90684623938625675866202159167, 14.23685106993605374935302924443, 15.58279307226094780146358295160, 16.53226705867431428965128939901, 17.466493076217919590926727839588, 18.47847686859920185352282822091, 19.29328024675900030369318376583, 20.43283084187637904728469228836, 21.422660758713762532719947581700, 21.91057787331782625323003807875, 23.424946278249000933868133724499, 24.01433777210052937439071232539, 24.82034868431729862662822713176, 26.05973381689535319270402376075

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