L-function 1-17-17.8-r0-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	17	$1$
good	2	$1 - iT$
	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	$1 + (0.707 + 0.707i)T$

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