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Properties

Label 1-2665-2665.1582-r0-0-0
Degree $1$
Conductor $2665$
Sign $0.908 - 0.417i$
Analytic cond. $12.3762$
Root an. cond. $12.3762$
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  + (−0.669 + 0.743i)2-s + (−0.258 + 0.965i)3-s + (−0.104 − 0.994i)4-s + (−0.544 − 0.838i)6-s + (−0.0523 + 0.998i)7-s + (0.809 + 0.587i)8-s + (−0.866 − 0.5i)9-s + (−0.629 − 0.777i)11-s + (0.987 + 0.156i)12-s + (−0.707 − 0.707i)14-s + (−0.978 + 0.207i)16-s + (−0.358 + 0.933i)17-s + (0.951 − 0.309i)18-s + (0.0523 − 0.998i)19-s + (−0.951 − 0.309i)21-s + (0.998 + 0.0523i)22-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (−0.258 + 0.965i)3-s + (−0.104 − 0.994i)4-s + (−0.544 − 0.838i)6-s + (−0.0523 + 0.998i)7-s + (0.809 + 0.587i)8-s + (−0.866 − 0.5i)9-s + (−0.629 − 0.777i)11-s + (0.987 + 0.156i)12-s + (−0.707 − 0.707i)14-s + (−0.978 + 0.207i)16-s + (−0.358 + 0.933i)17-s + (0.951 − 0.309i)18-s + (0.0523 − 0.998i)19-s + (−0.951 − 0.309i)21-s + (0.998 + 0.0523i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2665 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2665 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2665\)    =    \(5 \cdot 13 \cdot 41\)
Sign: $0.908 - 0.417i$
Analytic conductor: \(12.3762\)
Root analytic conductor: \(12.3762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2665} (1582, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2665,\ (0:\ ),\ 0.908 - 0.417i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2041502824 - 0.04461433307i\)
\(L(\frac12)\) \(\approx\) \(0.2041502824 - 0.04461433307i\)
\(L(1)\) \(\approx\) \(0.4385718736 + 0.3283114141i\)
\(L(1)\) \(\approx\) \(0.4385718736 + 0.3283114141i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.669 + 0.743i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
7 \( 1 + (-0.0523 + 0.998i)T \)
11 \( 1 + (-0.629 - 0.777i)T \)
17 \( 1 + (-0.358 + 0.933i)T \)
19 \( 1 + (0.0523 - 0.998i)T \)
23 \( 1 + (0.743 + 0.669i)T \)
29 \( 1 + (-0.933 + 0.358i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.406 + 0.913i)T \)
43 \( 1 + (-0.978 - 0.207i)T \)
47 \( 1 + (-0.891 - 0.453i)T \)
53 \( 1 + (-0.987 - 0.156i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
61 \( 1 + (-0.743 + 0.669i)T \)
67 \( 1 + (-0.933 + 0.358i)T \)
71 \( 1 + (0.358 - 0.933i)T \)
73 \( 1 + T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.544 - 0.838i)T \)
97 \( 1 + (-0.777 - 0.629i)T \)
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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

−19.20835512052464874374153013530, −18.72440450226564513268979529554, −17.95917306972369163576856777945, −17.53242366771559497869957776198, −16.69877251795564622249187419098, −16.30867024767050841025053032465, −15.10081156848020201594776260869, −14.02067179900325090269621604172, −13.48991620451858348667789151262, −12.7439966289879429707405691332, −12.315234821011854496874319859680, −11.29898367626442197001145553349, −10.92041535712604906890023814010, −10.00109351422243921010822499625, −9.42454565647295814611066175964, −8.24874681290336031769736572424, −7.78339479863710204467042771561, −7.09560836493710094475454764539, −6.50912883260220450907652212821, −5.19750801137312065568199830650, −4.41292747792219475282639139528, −3.39213196796528011623833033815, −2.49953305424136334092579746841, −1.768235506866843339614924534204, −0.85905071045018038641498088118, 0.10736740314677472267781271124, 1.54417586942405560425504994705, 2.71865657272731390349348427723, 3.52280860208540064845899559926, 4.914009712940184916197745798728, 5.13484325629357624014062286812, 6.066309495481361072064431633340, 6.58875713131283780686766585736, 7.77981397287930522952426023977, 8.70223095997514710543231235604, 8.86826412344211438465399321229, 9.79379176705973972609678859163, 10.50585605437765544834796296601, 11.18528555444576147066596255668, 11.76113469211282030699715172557, 13.03093826614993360310342800184, 13.70689599218513292731009082223, 14.81268592627845247433795888816, 15.193454388143402656905132828018, 15.722223783612462961516756658248, 16.377563196529203364004496786340, 17.09007895931923437316485726436, 17.735478938356149580123973669335, 18.40153313030131605791892537317, 19.24206661756819082590028970885

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