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Properties

Label 1-1617-1617.635-r1-0-0
Degree $1$
Conductor $1617$
Sign $0.605 + 0.795i$
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  + (0.791 − 0.611i)2-s + (0.251 − 0.967i)4-s + (0.575 + 0.817i)5-s + (−0.393 − 0.919i)8-s + (0.955 + 0.294i)10-s + (0.134 + 0.990i)13-s + (−0.873 − 0.486i)16-s + (0.447 + 0.894i)17-s + (0.913 + 0.406i)19-s + (0.936 − 0.351i)20-s + (0.988 + 0.149i)23-s + (−0.337 + 0.941i)25-s + (0.712 + 0.701i)26-s + (−0.0448 − 0.998i)29-s + (−0.669 − 0.743i)31-s + (−0.988 + 0.149i)32-s + ⋯
L(s)  = 1  + (0.791 − 0.611i)2-s + (0.251 − 0.967i)4-s + (0.575 + 0.817i)5-s + (−0.393 − 0.919i)8-s + (0.955 + 0.294i)10-s + (0.134 + 0.990i)13-s + (−0.873 − 0.486i)16-s + (0.447 + 0.894i)17-s + (0.913 + 0.406i)19-s + (0.936 − 0.351i)20-s + (0.988 + 0.149i)23-s + (−0.337 + 0.941i)25-s + (0.712 + 0.701i)26-s + (−0.0448 − 0.998i)29-s + (−0.669 − 0.743i)31-s + (−0.988 + 0.149i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(173.770\)
Root analytic conductor: \(173.770\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (635, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1617,\ (1:\ ),\ 0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.067488713 + 1.520002064i\)
\(L(\frac12)\) \(\approx\) \(3.067488713 + 1.520002064i\)
\(L(1)\) \(\approx\) \(1.761368500 - 0.1673454118i\)
\(L(1)\) \(\approx\) \(1.761368500 - 0.1673454118i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.791 - 0.611i)T \)
5 \( 1 + (0.575 + 0.817i)T \)
13 \( 1 + (0.134 + 0.990i)T \)
17 \( 1 + (0.447 + 0.894i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (0.988 + 0.149i)T \)
29 \( 1 + (-0.0448 - 0.998i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.842 + 0.538i)T \)
41 \( 1 + (0.393 + 0.919i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
47 \( 1 + (-0.999 + 0.0299i)T \)
53 \( 1 + (-0.998 + 0.0598i)T \)
59 \( 1 + (-0.599 - 0.800i)T \)
61 \( 1 + (0.998 + 0.0598i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.936 - 0.351i)T \)
73 \( 1 + (-0.999 - 0.0299i)T \)
79 \( 1 + (0.978 - 0.207i)T \)
83 \( 1 + (-0.134 + 0.990i)T \)
89 \( 1 + (0.0747 + 0.997i)T \)
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

−20.50649507710115062907814267560, −19.69598537749530011843524375237, −18.3724983935091109509541361149, −17.695984996359884415496884015630, −17.12954085062236281034942210406, −16.11592638912104342066652264925, −15.929611711027647908625808388049, −14.82608919359758468191905880943, −14.08308707607552075550062684207, −13.44966538929944211347577745079, −12.69802770499580057414601970609, −12.211971625993558891230555138852, −11.20879987511283490719053715592, −10.27561542836672848556493381574, −9.12471121520661046231174232746, −8.6960218408232678675091310205, −7.5657810183323790231467969666, −7.02496544715516819529841170995, −5.85696424637837371339895444260, −5.235755805795934280659738137683, −4.79830347608462674130709082536, −3.48402102057124895074203205289, −2.85406351021288031363738348152, −1.60410236391178342274562729260, −0.43786282942570812961923001366, 1.24310495858400453152716774326, 1.92262973198518531385456402548, 2.94291451519391241027087307409, 3.607624570454886289816434571155, 4.56637280728579293599290037954, 5.557364141989918656311576760731, 6.23362255657842212066684032150, 6.91735285762467730394660533255, 7.93107386718129378565116846992, 9.35430872553727468669994346662, 9.7434256794084261891104323282, 10.6679860628442271831265257521, 11.326454869415260006359883438724, 11.95156644083691667257610967763, 13.04227401642771355585605299877, 13.50451909320511227293901598816, 14.491910059826619152983070993631, 14.69013369943209343660640665206, 15.69658458266978916849681523631, 16.58968077324940610171438118496, 17.50578576425894083081526225656, 18.41560486174610510656343112873, 19.03155293184893776333923985758, 19.498288594505058188551261940960, 20.77377311060753509597406434239

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