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Properties

Label 1-340-340.159-r0-0-0
Degree $1$
Conductor $340$
Sign $0.978 + 0.204i$
Analytic cond. $1.57895$
Root an. cond. $1.57895$
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  + (0.923 − 0.382i)3-s + (−0.382 + 0.923i)7-s + (0.707 − 0.707i)9-s + (0.923 + 0.382i)11-s + i·13-s + (−0.707 − 0.707i)19-s + i·21-s + (0.923 + 0.382i)23-s + (0.382 − 0.923i)27-s + (0.382 + 0.923i)29-s + (0.923 − 0.382i)31-s + 33-s + (−0.923 + 0.382i)37-s + (0.382 + 0.923i)39-s + (−0.382 + 0.923i)41-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)3-s + (−0.382 + 0.923i)7-s + (0.707 − 0.707i)9-s + (0.923 + 0.382i)11-s + i·13-s + (−0.707 − 0.707i)19-s + i·21-s + (0.923 + 0.382i)23-s + (0.382 − 0.923i)27-s + (0.382 + 0.923i)29-s + (0.923 − 0.382i)31-s + 33-s + (−0.923 + 0.382i)37-s + (0.382 + 0.923i)39-s + (−0.382 + 0.923i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 340 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(340\)    =    \(2^{2} \cdot 5 \cdot 17\)
Sign: $0.978 + 0.204i$
Analytic conductor: \(1.57895\)
Root analytic conductor: \(1.57895\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{340} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 340,\ (0:\ ),\ 0.978 + 0.204i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.764218563 + 0.1827715834i\)
\(L(\frac12)\) \(\approx\) \(1.764218563 + 0.1827715834i\)
\(L(1)\) \(\approx\) \(1.434005410 + 0.03342876665i\)
\(L(1)\) \(\approx\) \(1.434005410 + 0.03342876665i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 + (-0.382 + 0.923i)T \)
11 \( 1 + (0.923 + 0.382i)T \)
13 \( 1 + iT \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (0.923 + 0.382i)T \)
29 \( 1 + (0.382 + 0.923i)T \)
31 \( 1 + (0.923 - 0.382i)T \)
37 \( 1 + (-0.923 + 0.382i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 - 0.707i)T \)
59 \( 1 + (0.707 - 0.707i)T \)
61 \( 1 + (0.382 - 0.923i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.923 + 0.382i)T \)
73 \( 1 + (0.382 + 0.923i)T \)
79 \( 1 + (0.923 + 0.382i)T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.382 - 0.923i)T \)
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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

−25.07798980523185107530339390088, −24.220087739867903803195162624702, −22.92591804481626203864333467612, −22.3675388487219717765602270125, −21.0486557113181544563812095716, −20.59438526853961230560773712795, −19.3421833570074115551899584426, −19.2722440461191746874614485992, −17.607955369379767529893268358958, −16.77650034143472931859587988580, −15.85156855038104466729992848787, −14.88789853176910201448134573633, −14.0647024557049882602137197111, −13.29777230320144513722630968421, −12.31267200688217858286338491849, −10.78668231066133381271664998628, −10.190601480774163757617940835385, −9.12318520351370451171668794647, −8.21448050741439984703364276263, −7.24696885531006422401959136720, −6.127062460821146437773432539549, −4.57527837880792827415321689310, −3.711716651143524207631467010218, −2.78316797047262418981969863639, −1.18935530254635397390042702761, 1.54245978709635681508077759484, 2.555346647276173710224998639442, 3.660540833372388161776754301004, 4.854026279332052446085053349467, 6.48242135963234809456828897953, 6.97930570713167784269725682396, 8.49149826302369917252626139160, 9.04894401106156007218806428949, 9.86999984789496817584540285497, 11.45687742181360656356301399320, 12.29394977921357097981543361342, 13.163666722457816236068125538963, 14.16722515139057850899321431206, 14.97888199201727510501333144823, 15.71594601862524413730502587415, 16.95839457738785121161250279436, 17.969270929452004849894473688939, 19.15706353312173247858497859010, 19.27423985224272768846575370956, 20.47924877347322878795205093048, 21.42738879964405062045427186402, 22.120624717454064647525079170719, 23.35072665891277633344375156531, 24.25317794443426664368712724445, 25.10954558936880014006682390753

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