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Properties

Label 1-1617-1617.689-r1-0-0
Degree $1$
Conductor $1617$
Sign $0.221 - 0.975i$
Analytic cond. $173.770$
Root an. cond. $173.770$
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  + (0.420 − 0.907i)2-s + (−0.646 − 0.762i)4-s + (0.998 − 0.0598i)5-s + (−0.963 + 0.266i)8-s + (0.365 − 0.930i)10-s + (−0.995 + 0.0896i)13-s + (−0.163 + 0.986i)16-s + (−0.791 + 0.611i)17-s + (0.913 − 0.406i)19-s + (−0.691 − 0.722i)20-s + (−0.826 + 0.563i)23-s + (0.992 − 0.119i)25-s + (−0.337 + 0.941i)26-s + (0.473 − 0.880i)29-s + (−0.669 + 0.743i)31-s + (0.826 + 0.563i)32-s + ⋯
L(s)  = 1  + (0.420 − 0.907i)2-s + (−0.646 − 0.762i)4-s + (0.998 − 0.0598i)5-s + (−0.963 + 0.266i)8-s + (0.365 − 0.930i)10-s + (−0.995 + 0.0896i)13-s + (−0.163 + 0.986i)16-s + (−0.791 + 0.611i)17-s + (0.913 − 0.406i)19-s + (−0.691 − 0.722i)20-s + (−0.826 + 0.563i)23-s + (0.992 − 0.119i)25-s + (−0.337 + 0.941i)26-s + (0.473 − 0.880i)29-s + (−0.669 + 0.743i)31-s + (0.826 + 0.563i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.221 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.221 - 0.975i$
Analytic conductor: \(173.770\)
Root analytic conductor: \(173.770\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1617,\ (1:\ ),\ 0.221 - 0.975i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.096593103 - 1.673941975i\)
\(L(\frac12)\) \(\approx\) \(2.096593103 - 1.673941975i\)
\(L(1)\) \(\approx\) \(1.193999750 - 0.6561195210i\)
\(L(1)\) \(\approx\) \(1.193999750 - 0.6561195210i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.420 - 0.907i)T \)
5 \( 1 + (0.998 - 0.0598i)T \)
13 \( 1 + (-0.995 + 0.0896i)T \)
17 \( 1 + (-0.791 + 0.611i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (-0.826 + 0.563i)T \)
29 \( 1 + (0.473 - 0.880i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (0.525 + 0.850i)T \)
41 \( 1 + (0.963 - 0.266i)T \)
43 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (0.193 + 0.981i)T \)
53 \( 1 + (0.925 - 0.379i)T \)
59 \( 1 + (0.251 + 0.967i)T \)
61 \( 1 + (-0.925 - 0.379i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.691 - 0.722i)T \)
73 \( 1 + (0.193 - 0.981i)T \)
79 \( 1 + (0.978 + 0.207i)T \)
83 \( 1 + (0.995 + 0.0896i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−20.49115911163897436537995880887, −19.85619530017086535343532576802, −18.46924839264055824358673283056, −18.07341382692052243460585932096, −17.4335883253883601628847736192, −16.525203450234195256523362352222, −16.14067280853419490347634279973, −15.01583540240618368812860445458, −14.41471291108688951617239805526, −13.84310973954220841238415133691, −13.05527818118596691625987650145, −12.39564566780961751579481007936, −11.50714052922063442231921618837, −10.31843975012710289569150802021, −9.53479871304092470531348819500, −8.98395648509001910117675160716, −7.90799825963190384558185446761, −7.15804274507798921512277652277, −6.418083474400422124850945794233, −5.57733201355350774986439815345, −4.97765876747955478280426021628, −4.064982171000101035145556765517, −2.89488990395465978265962556673, −2.15011171558347729841404358683, −0.63437324154185315603790552666, 0.63695506285770172258087225675, 1.751428400578037996002037065801, 2.368677651444206939599057260896, 3.27022999079411115722499189092, 4.382902669644034157681899721306, 5.090197664459886147590867475185, 5.882603878445859534609595728959, 6.65920764907905691443106385201, 7.8440686817653050167171906586, 9.01940486815146189151648908523, 9.51627304683174229886563798613, 10.25095672762203269526453634449, 10.9318234949697155605353203344, 11.93383897130712582761670280151, 12.4638419173248115197645264211, 13.53033320320044256619473040462, 13.718577142582900892659271192536, 14.694326283947666880305199105355, 15.35404595559189102293493270883, 16.463787598952461474154116110281, 17.51545627446499902211024032420, 17.82022870434116364294371582226, 18.667175171627139128138193928455, 19.722740607375114432453582469938, 19.955112757597708189800915163822

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