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Properties

Label 1-647-647.218-r0-0-0
Degree $1$
Conductor $647$
Sign $0.676 + 0.736i$
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  + (0.0922 + 0.995i)2-s + (0.445 − 0.895i)3-s + (−0.982 + 0.183i)4-s + (−0.602 − 0.798i)5-s + (0.932 + 0.361i)6-s + (−0.273 + 0.961i)7-s + (−0.273 − 0.961i)8-s + (−0.602 − 0.798i)9-s + (0.739 − 0.673i)10-s + (−0.602 + 0.798i)11-s + (−0.273 + 0.961i)12-s + (−0.273 + 0.961i)13-s + (−0.982 − 0.183i)14-s + (−0.982 + 0.183i)15-s + (0.932 − 0.361i)16-s + (0.445 − 0.895i)17-s + ⋯
L(s)  = 1  + (0.0922 + 0.995i)2-s + (0.445 − 0.895i)3-s + (−0.982 + 0.183i)4-s + (−0.602 − 0.798i)5-s + (0.932 + 0.361i)6-s + (−0.273 + 0.961i)7-s + (−0.273 − 0.961i)8-s + (−0.602 − 0.798i)9-s + (0.739 − 0.673i)10-s + (−0.602 + 0.798i)11-s + (−0.273 + 0.961i)12-s + (−0.273 + 0.961i)13-s + (−0.982 − 0.183i)14-s + (−0.982 + 0.183i)15-s + (0.932 − 0.361i)16-s + (0.445 − 0.895i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 647 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(647\)
Sign: $0.676 + 0.736i$
Analytic conductor: \(3.00465\)
Root analytic conductor: \(3.00465\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{647} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 647,\ (0:\ ),\ 0.676 + 0.736i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.079133030 + 0.4738453248i\)
\(L(\frac12)\) \(\approx\) \(1.079133030 + 0.4738453248i\)
\(L(1)\) \(\approx\) \(0.9602132841 + 0.2427995486i\)
\(L(1)\) \(\approx\) \(0.9602132841 + 0.2427995486i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad647 \( 1 \)
good2 \( 1 + (0.0922 + 0.995i)T \)
3 \( 1 + (0.445 - 0.895i)T \)
5 \( 1 + (-0.602 - 0.798i)T \)
7 \( 1 + (-0.273 + 0.961i)T \)
11 \( 1 + (-0.602 + 0.798i)T \)
13 \( 1 + (-0.273 + 0.961i)T \)
17 \( 1 + (0.445 - 0.895i)T \)
19 \( 1 + (0.932 - 0.361i)T \)
23 \( 1 + (0.932 + 0.361i)T \)
29 \( 1 + (0.932 + 0.361i)T \)
31 \( 1 + (-0.602 + 0.798i)T \)
37 \( 1 + (0.932 + 0.361i)T \)
41 \( 1 + (-0.273 - 0.961i)T \)
43 \( 1 + (0.0922 - 0.995i)T \)
47 \( 1 + (0.445 - 0.895i)T \)
53 \( 1 + (0.932 + 0.361i)T \)
59 \( 1 + (0.739 - 0.673i)T \)
61 \( 1 + (0.445 + 0.895i)T \)
67 \( 1 + (0.445 + 0.895i)T \)
71 \( 1 + (0.932 + 0.361i)T \)
73 \( 1 + (0.0922 + 0.995i)T \)
79 \( 1 + (0.0922 - 0.995i)T \)
83 \( 1 + (0.739 + 0.673i)T \)
89 \( 1 + (0.0922 + 0.995i)T \)
97 \( 1 + (0.739 - 0.673i)T \)
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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

−22.751730791282841638304267710098, −21.815293191641030685212876939253, −21.11894758400117512113446012605, −20.235269977798348325291489597614, −19.67592362301413261226913421926, −18.994804399857887609557586647421, −18.06577137697411636492299403799, −16.96002089865195808733955733860, −16.10685352428039919598865389978, −15.0324734213759676173892713312, −14.43755145395691217145031789004, −13.56620870401325600602435276363, −12.7745789015342337937719979797, −11.45220923211132612534770118847, −10.773045119836994170960104231654, −10.28542482263714980107763227421, −9.51978095009309633547386158297, −8.133280896079664360635105445282, −7.766666285010033805626412271326, −6.033117136847019738565729565949, −4.93266763641699864463028975274, −3.90200011644413750002307011254, −3.26438660431886569277617090915, −2.64129438779228519979209005606, −0.73957997239233705588816265844, 0.9454861460713770536787276105, 2.45999068908803718116748379812, 3.60654481584308096731646065442, 4.97360333621720316324219356231, 5.47674271829200332350111967726, 6.99272189956624265820706301643, 7.25291587192985242423180805428, 8.39918925038692263158299455398, 9.01541086273432671494865663693, 9.686777478467923275358299656, 11.75240485201239064646254087561, 12.23445101520150206312316748786, 13.01134186607887983198483712228, 13.81433354190546041040519322080, 14.756577175589542060663471015012, 15.5561858809720539112347416925, 16.19155585259453740247855634588, 17.18386902103226649463087001906, 18.12758220959998586213840440614, 18.6999344952960425475280921520, 19.489896534602197653601210032108, 20.44337092358608064016150147678, 21.39093099691859746033853160559, 22.45739470899957978382359712609, 23.498776040175251623937799249957

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