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Properties

Label 1-896-896.125-r1-0-0
Degree $1$
Conductor $896$
Sign $0.740 + 0.671i$
Analytic cond. $96.2885$
Root an. cond. $96.2885$
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  + (0.195 − 0.980i)3-s + (−0.831 − 0.555i)5-s + (−0.923 − 0.382i)9-s + (0.980 − 0.195i)11-s + (0.831 − 0.555i)13-s + (−0.707 + 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.555 − 0.831i)19-s + (−0.382 + 0.923i)23-s + (0.382 + 0.923i)25-s + (−0.555 + 0.831i)27-s + (−0.980 − 0.195i)29-s − i·31-s − i·33-s + (−0.555 + 0.831i)37-s + ⋯
L(s)  = 1  + (0.195 − 0.980i)3-s + (−0.831 − 0.555i)5-s + (−0.923 − 0.382i)9-s + (0.980 − 0.195i)11-s + (0.831 − 0.555i)13-s + (−0.707 + 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.555 − 0.831i)19-s + (−0.382 + 0.923i)23-s + (0.382 + 0.923i)25-s + (−0.555 + 0.831i)27-s + (−0.980 − 0.195i)29-s − i·31-s − i·33-s + (−0.555 + 0.831i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.740 + 0.671i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ 0.740 + 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8285781450 + 0.3196178600i\)
\(L(\frac12)\) \(\approx\) \(0.8285781450 + 0.3196178600i\)
\(L(1)\) \(\approx\) \(0.8568760901 - 0.3094166501i\)
\(L(1)\) \(\approx\) \(0.8568760901 - 0.3094166501i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.195 - 0.980i)T \)
5 \( 1 + (-0.831 - 0.555i)T \)
11 \( 1 + (0.980 - 0.195i)T \)
13 \( 1 + (0.831 - 0.555i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (-0.555 - 0.831i)T \)
23 \( 1 + (-0.382 + 0.923i)T \)
29 \( 1 + (-0.980 - 0.195i)T \)
31 \( 1 - iT \)
37 \( 1 + (-0.555 + 0.831i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
43 \( 1 + (-0.195 - 0.980i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (-0.980 + 0.195i)T \)
59 \( 1 + (0.831 + 0.555i)T \)
61 \( 1 + (-0.195 + 0.980i)T \)
67 \( 1 + (0.195 - 0.980i)T \)
71 \( 1 + (-0.923 + 0.382i)T \)
73 \( 1 + (-0.923 - 0.382i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (0.555 + 0.831i)T \)
89 \( 1 + (0.382 + 0.923i)T \)
97 \( 1 - iT \)
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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

−21.7230354024039738614794225411, −20.67052173366324698417068852274, −20.370224434883331867136920096060, −19.149128114579333085933525216036, −18.83227308562045948958398522787, −17.62225228369095315485935934927, −16.47534744264150208002549378192, −16.255236889521839221541385137, −15.21026124444375906237257592028, −14.454572660857623487573583841625, −14.11262001881972652794178227487, −12.65058429968397381171055398947, −11.64885547288789529021708222269, −11.17629626129106313999375730291, −10.250049624665941331157944450485, −9.411849839117530210778103484884, −8.56435926301970719457148258610, −7.74781698772041383234422858440, −6.65134745663456372910092002036, −5.78467118003005087830628687781, −4.47315769726659629474701130357, −3.88292289938094799605570260210, −3.17584376179252967400927540818, −1.86690918287981802560402709995, −0.21241085302444891619634115906, 0.99895592119066257601685632279, 1.67418112598566283626248201036, 3.255457237416450236372153876461, 3.80678916954218947682392074068, 5.16305365129238612130225287546, 6.13404369262557044215576113776, 6.97570100929817081909111294842, 7.92271542810683035446664014167, 8.526631218297416879860847935986, 9.24169984608094626502952742353, 10.65293222675964203025729297086, 11.62197270908304679777798753236, 12.07440847418120412767256606469, 13.05475514435742618258846816054, 13.56397156574952254439675076618, 14.71143844226099120475169521871, 15.315042162752327242410480792023, 16.40621407929788441160683968200, 17.14293534089002322983227030330, 17.881777310942835863322068773572, 18.92344976053452098588699165495, 19.43918984113489908754962679636, 20.06494863221002974865084722100, 20.82796799352369670223023280972, 21.92206566030065998516922794736

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