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Properties

Label 1-1568-1568.701-r0-0-0
Degree $1$
Conductor $1568$
Sign $0.922 + 0.386i$
Analytic cond. $7.28176$
Root an. cond. $7.28176$
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  + (0.532 − 0.846i)3-s + (0.846 + 0.532i)5-s + (−0.433 − 0.900i)9-s + (−0.943 + 0.330i)11-s + (0.330 + 0.943i)13-s + (0.900 − 0.433i)15-s + (−0.623 + 0.781i)17-s + (−0.707 − 0.707i)19-s + (−0.781 + 0.623i)23-s + (0.433 + 0.900i)25-s + (−0.993 − 0.111i)27-s + (0.993 − 0.111i)29-s + 31-s + (−0.222 + 0.974i)33-s + (0.111 + 0.993i)37-s + ⋯
L(s)  = 1  + (0.532 − 0.846i)3-s + (0.846 + 0.532i)5-s + (−0.433 − 0.900i)9-s + (−0.943 + 0.330i)11-s + (0.330 + 0.943i)13-s + (0.900 − 0.433i)15-s + (−0.623 + 0.781i)17-s + (−0.707 − 0.707i)19-s + (−0.781 + 0.623i)23-s + (0.433 + 0.900i)25-s + (−0.993 − 0.111i)27-s + (0.993 − 0.111i)29-s + 31-s + (−0.222 + 0.974i)33-s + (0.111 + 0.993i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.922 + 0.386i$
Analytic conductor: \(7.28176\)
Root analytic conductor: \(7.28176\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1568,\ (0:\ ),\ 0.922 + 0.386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.909368155 + 0.3837754394i\)
\(L(\frac12)\) \(\approx\) \(1.909368155 + 0.3837754394i\)
\(L(1)\) \(\approx\) \(1.343626575 - 0.05131489868i\)
\(L(1)\) \(\approx\) \(1.343626575 - 0.05131489868i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.532 - 0.846i)T \)
5 \( 1 + (0.846 + 0.532i)T \)
11 \( 1 + (-0.943 + 0.330i)T \)
13 \( 1 + (0.330 + 0.943i)T \)
17 \( 1 + (-0.623 + 0.781i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
23 \( 1 + (-0.781 + 0.623i)T \)
29 \( 1 + (0.993 - 0.111i)T \)
31 \( 1 + T \)
37 \( 1 + (0.111 + 0.993i)T \)
41 \( 1 + (0.974 - 0.222i)T \)
43 \( 1 + (0.532 + 0.846i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (0.993 + 0.111i)T \)
59 \( 1 + (-0.532 - 0.846i)T \)
61 \( 1 + (0.993 - 0.111i)T \)
67 \( 1 + (0.707 + 0.707i)T \)
71 \( 1 + (0.781 - 0.623i)T \)
73 \( 1 + (-0.433 - 0.900i)T \)
79 \( 1 - T \)
83 \( 1 + (0.943 + 0.330i)T \)
89 \( 1 + (0.433 + 0.900i)T \)
97 \( 1 + T \)
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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

−20.461847462829315841099079900476, −20.10414251784040505125901406198, −19.03164866025662240052382686833, −18.12985767798249035077533478434, −17.50919921236161696372607575361, −16.56618632263936027098778421809, −15.93172161962129168453712703508, −15.42095172372381283044433709645, −14.31265129045718414312483942888, −13.80627717094003892002569265553, −13.06408292466834437619652438511, −12.316961125804520126932671547086, −11.0487133295764232491317540704, −10.29211720641295819549199617143, −9.97229842613381696857249399891, −8.75992458111598137954590721787, −8.48445708632071165108462610116, −7.52382926068020486031616886713, −6.13598614857001568843353486969, −5.52371859428510086668116054278, −4.71747002530724891974102439899, −3.91010648287862811022383014997, −2.65759874975782285099971957469, −2.29099817951361306840593109591, −0.687320627933075490070016648504, 1.18689627028265438375264749025, 2.26703531635893468222573796912, 2.53800832491769896880703271745, 3.8031728531950760928167899236, 4.840021696812674788824278237207, 6.15033651060093089293023674014, 6.42404401963669796193535069050, 7.36008302508283098419259995629, 8.21281680984393025522643812285, 8.97933675205578779208510330038, 9.79976534005659984472585229489, 10.63224403218432444551688249306, 11.46705335754949817466817739953, 12.432841564458385716803843912860, 13.22988366737954967981289824884, 13.67777823445500575249721390352, 14.38642542558968730395250598195, 15.21381173034988080647436095004, 15.91154675250396929346529759143, 17.286855818174296811046017718668, 17.601555990346381658772665166998, 18.3548635318690649167609628207, 19.065479627416045937263304901, 19.62928333669217385215922958558, 20.61685525039966110530217382335

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