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Properties

Label 1-1260-1260.419-r1-0-0
Degree $1$
Conductor $1260$
Sign $0.984 + 0.173i$
Analytic cond. $135.405$
Root an. cond. $135.405$
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  + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s − 17-s + 19-s + (0.5 + 0.866i)23-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s − 37-s + (−0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s + (−0.5 + 0.866i)47-s + 53-s + (0.5 + 0.866i)59-s + (0.5 − 0.866i)61-s + (−0.5 − 0.866i)67-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s − 17-s + 19-s + (0.5 + 0.866i)23-s + (0.5 − 0.866i)29-s + (−0.5 − 0.866i)31-s − 37-s + (−0.5 − 0.866i)41-s + (−0.5 + 0.866i)43-s + (−0.5 + 0.866i)47-s + 53-s + (0.5 + 0.866i)59-s + (0.5 − 0.866i)61-s + (−0.5 − 0.866i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(135.405\)
Root analytic conductor: \(135.405\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1260,\ (1:\ ),\ 0.984 + 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.635439583 + 0.1430824234i\)
\(L(\frac12)\) \(\approx\) \(1.635439583 + 0.1430824234i\)
\(L(1)\) \(\approx\) \(0.9853766617 + 0.01808965921i\)
\(L(1)\) \(\approx\) \(0.9853766617 + 0.01808965921i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 - T \)
19 \( 1 + T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + T \)
59 \( 1 + (0.5 + 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−20.88397670485997262624961032644, −20.0268614811691506875877605990, −19.371784063984381825452606298804, −18.47628033324501171245937556996, −17.97119518300367685794969436649, −16.87575404499368356581509194951, −16.29860929858467259709310449082, −15.578561494874120145374732449835, −14.602917115858608326746665595539, −13.88961678794936692508353667127, −13.20819715752872793180025513328, −12.25340852539949882104520467153, −11.48395231937760128372610046784, −10.72179708867698488009612195468, −9.90942597346777373488567316602, −8.838884259363994808924574642832, −8.425433554983009628481721079062, −7.09621485637432371390437368561, −6.68608968705333565892134985707, −5.408312169880550565911572541675, −4.820672001435643573412932879024, −3.66142017327993150146192447043, −2.7912887757751360982890095563, −1.76934631042559470591631713786, −0.533687935914951868101571420581, 0.58267661355898346113147962556, 1.89477676945449909100117436879, 2.75507756392593482432315252813, 3.77135877874809860421948670309, 4.88137242416879217554337246804, 5.43427704700173221605969143721, 6.60707158811128376831402145284, 7.45037672072850858511309600257, 8.06400738196263952724698653715, 9.22977827414667474889664691236, 9.86360769545699096905947054122, 10.68493957292116642361258053064, 11.57873658895521130740296993204, 12.37584194348713881284296410500, 13.19269360227619485181829734661, 13.79444047989981323397950582999, 15.010137621123032938985708265276, 15.36924514472383547689186671109, 16.183534962535250778402489570021, 17.35185834857379452651344137547, 17.69403657689106890290275042543, 18.51431458628700471399571622668, 19.51752206484510543345063346585, 20.123521180857256893482895054530, 20.79642208711211053179828409860

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