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Properties

Label 1-647-647.2-r0-0-0
Degree $1$
Conductor $647$
Sign $-0.309 - 0.951i$
Analytic cond. $3.00465$
Root an. cond. $3.00465$
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  + (−0.618 − 0.786i)2-s + (−0.994 − 0.106i)3-s + (−0.236 + 0.971i)4-s + (0.0340 + 0.999i)5-s + (0.530 + 0.847i)6-s + (0.0145 − 0.999i)7-s + (0.909 − 0.415i)8-s + (0.977 + 0.212i)9-s + (0.764 − 0.644i)10-s + (−0.505 + 0.862i)11-s + (0.338 − 0.940i)12-s + (−0.745 + 0.666i)13-s + (−0.795 + 0.606i)14-s + (0.0728 − 0.997i)15-s + (−0.888 − 0.458i)16-s + (−0.454 − 0.890i)17-s + ⋯
L(s)  = 1  + (−0.618 − 0.786i)2-s + (−0.994 − 0.106i)3-s + (−0.236 + 0.971i)4-s + (0.0340 + 0.999i)5-s + (0.530 + 0.847i)6-s + (0.0145 − 0.999i)7-s + (0.909 − 0.415i)8-s + (0.977 + 0.212i)9-s + (0.764 − 0.644i)10-s + (−0.505 + 0.862i)11-s + (0.338 − 0.940i)12-s + (−0.745 + 0.666i)13-s + (−0.795 + 0.606i)14-s + (0.0728 − 0.997i)15-s + (−0.888 − 0.458i)16-s + (−0.454 − 0.890i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.309 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.309 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(647\)
Sign: $-0.309 - 0.951i$
Analytic conductor: \(3.00465\)
Root analytic conductor: \(3.00465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{647} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 647,\ (0:\ ),\ -0.309 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2367148389 - 0.3258331712i\)
\(L(\frac12)\) \(\approx\) \(0.2367148389 - 0.3258331712i\)
\(L(1)\) \(\approx\) \(0.4708154742 - 0.1484341253i\)
\(L(1)\) \(\approx\) \(0.4708154742 - 0.1484341253i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad647 \( 1 \)
good2 \( 1 + (-0.618 - 0.786i)T \)
3 \( 1 + (-0.994 - 0.106i)T \)
5 \( 1 + (0.0340 + 0.999i)T \)
7 \( 1 + (0.0145 - 0.999i)T \)
11 \( 1 + (-0.505 + 0.862i)T \)
13 \( 1 + (-0.745 + 0.666i)T \)
17 \( 1 + (-0.454 - 0.890i)T \)
19 \( 1 + (-0.419 - 0.907i)T \)
23 \( 1 + (-0.989 + 0.145i)T \)
29 \( 1 + (0.939 + 0.343i)T \)
31 \( 1 + (0.993 + 0.116i)T \)
37 \( 1 + (0.999 + 0.0194i)T \)
41 \( 1 + (-0.965 + 0.259i)T \)
43 \( 1 + (0.739 - 0.673i)T \)
47 \( 1 + (0.925 - 0.379i)T \)
53 \( 1 + (-0.982 - 0.183i)T \)
59 \( 1 + (-0.365 + 0.930i)T \)
61 \( 1 + (0.751 + 0.659i)T \)
67 \( 1 + (-0.850 - 0.526i)T \)
71 \( 1 + (0.800 - 0.598i)T \)
73 \( 1 + (0.610 - 0.792i)T \)
79 \( 1 + (0.480 - 0.877i)T \)
83 \( 1 + (-0.0437 - 0.999i)T \)
89 \( 1 + (-0.329 - 0.944i)T \)
97 \( 1 + (-0.979 + 0.202i)T \)
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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

−23.46521598199724435521981626246, −22.325136987982842216745735871056, −21.67334486638800704572704527673, −20.7283545814493438200803554162, −19.53266987316806797062708061873, −18.82494831044124630137701899509, −17.93022127220217651815481699076, −17.274188129491834021441842031365, −16.58883819999635297225105384480, −15.739928319220487259456387491624, −15.366264740548619859390161771419, −14.0757182858277986862759999770, −12.86728930988741296557917636004, −12.28493414261015573920335717908, −11.21295598978161409464340132709, −10.208503390345277824805674088708, −9.54640117723050229576742414521, −8.29105827140029029820274634098, −8.03483409506651450722130363996, −6.35426918213582496246295649715, −5.85427914897370962465910913271, −5.13678730723089207710456296470, −4.23641925012230146053332864488, −2.21013829043089899881817430945, −0.932181248972089274410417648602, 0.35813569162990557804641852293, 1.8693977746280602404413243957, 2.778525168955153722956661437436, 4.23243911536913272038872549506, 4.76731893841588768523952497451, 6.54995500582944341770757863011, 7.12690378226675314614037779520, 7.77281435830269500575441987926, 9.42553321852151221012538074516, 10.199347184807199250993211608614, 10.65374623491838451197502161232, 11.57854435997813568652956578969, 12.16581561269380978382712147780, 13.320355307048880396128429463429, 13.9718629302907702404461924601, 15.37912010985386175883146033244, 16.26385230756191089860090234845, 17.2381720806755031234825894846, 17.75140527570892712354800037404, 18.34987552532153844566475592076, 19.32154645286062730432894991744, 20.009746077016534472454113973065, 21.044905394299749055370066017792, 21.917898428911007431860538486238, 22.44361118259731823562601953985

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