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Properties

Label 1-17-17.8-r0-0-0
Degree $1$
Conductor $17$
Sign $0.0758 - 0.997i$
Analytic cond. $0.0789476$
Root an. cond. $0.0789476$
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  − i·2-s + (−0.707 − 0.707i)3-s − 4-s + (0.707 + 0.707i)5-s + (−0.707 + 0.707i)6-s + (0.707 − 0.707i)7-s + i·8-s + i·9-s + (0.707 − 0.707i)10-s + (−0.707 + 0.707i)11-s + (0.707 + 0.707i)12-s − 13-s + (−0.707 − 0.707i)14-s − i·15-s + 16-s + ⋯
L(s)  = 1  − i·2-s + (−0.707 − 0.707i)3-s − 4-s + (0.707 + 0.707i)5-s + (−0.707 + 0.707i)6-s + (0.707 − 0.707i)7-s + i·8-s + i·9-s + (0.707 − 0.707i)10-s + (−0.707 + 0.707i)11-s + (0.707 + 0.707i)12-s − 13-s + (−0.707 − 0.707i)14-s − i·15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0758 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0758 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(17\)
Sign: $0.0758 - 0.997i$
Analytic conductor: \(0.0789476\)
Root analytic conductor: \(0.0789476\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 17,\ (0:\ ),\ 0.0758 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3930510725 - 0.3642743770i\)
\(L(\frac12)\) \(\approx\) \(0.3930510725 - 0.3642743770i\)
\(L(1)\) \(\approx\) \(0.6432504208 - 0.4152647231i\)
\(L(1)\) \(\approx\) \(0.6432504208 - 0.4152647231i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 - iT \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
11 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 - T \)
19 \( 1 - iT \)
23 \( 1 + (-0.707 + 0.707i)T \)
29 \( 1 + (0.707 + 0.707i)T \)
31 \( 1 + (-0.707 - 0.707i)T \)
37 \( 1 + (-0.707 - 0.707i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + iT \)
47 \( 1 - T \)
53 \( 1 - iT \)
59 \( 1 + iT \)
61 \( 1 + (0.707 - 0.707i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 - iT \)
89 \( 1 - T \)
97 \( 1 + (0.707 + 0.707i)T \)
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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

−41.78113281428913157097751556175, −40.49088600448141846299605349922, −39.74901620953299443468760304689, −37.60885502423026500222176839904, −36.36085380029876327740270208241, −34.590307070145623317103704703622, −33.88335533870203301031755919184, −32.54134894116512581757908671454, −31.567769782677363732134827762391, −29.06275433365868831471832760164, −27.82316850298224847048343793504, −26.64256881139864551522557469254, −24.9083980473007162621752234757, −23.84320748008473993086545937703, −22.09582685545340040570870347604, −21.10804943703809034169584922445, −18.25002335183789884203414139403, −17.075532607972307550969197780390, −15.91778375265039377644843496150, −14.36607509160395072561089268158, −12.36855757010477121158113533374, −10.01238724806638376954022051124, −8.45859216052355763456069731246, −5.87734309066374149436213082502, −4.851474204208871568224755857489, 2.12217010145779887966076947147, 5.075115308096543817057063202124, 7.40782305438915425283646460554, 10.099631006022388005816075471623, 11.27986990922035091305091711802, 12.937810668313017633046916861138, 14.258617677355082952028600362205, 17.43658391124540795162895968480, 18.05131796137856559804470653247, 19.70335533978972512386379496784, 21.44929725716774721606237700956, 22.69234679002922915638226316278, 23.9873787961286208002815348013, 26.12904035449572456747400044193, 27.72733025278830988965908446233, 29.163238467224250497882396208664, 29.965807102477822414522754471951, 31.04038193885368095363004612009, 33.20789636131639512568872683242, 34.46015772175810913043809216627, 36.3291400306786646872035852317, 36.97068255266554082689339828045, 38.74831002439893138928391708228, 39.89239477527859477534990061176, 41.06496375562159222630440514728

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