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Properties

Label 1-896-896.307-r0-0-0
Degree $1$
Conductor $896$
Sign $0.336 - 0.941i$
Analytic cond. $4.16100$
Root an. cond. $4.16100$
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  + (0.831 − 0.555i)3-s + (−0.980 + 0.195i)5-s + (0.382 − 0.923i)9-s + (0.555 − 0.831i)11-s + (0.980 + 0.195i)13-s + (−0.707 + 0.707i)15-s + (−0.707 − 0.707i)17-s + (−0.195 + 0.980i)19-s + (0.923 + 0.382i)23-s + (0.923 − 0.382i)25-s + (−0.195 − 0.980i)27-s + (0.555 + 0.831i)29-s − i·31-s − i·33-s + (0.195 + 0.980i)37-s + ⋯
L(s)  = 1  + (0.831 − 0.555i)3-s + (−0.980 + 0.195i)5-s + (0.382 − 0.923i)9-s + (0.555 − 0.831i)11-s + (0.980 + 0.195i)13-s + (−0.707 + 0.707i)15-s + (−0.707 − 0.707i)17-s + (−0.195 + 0.980i)19-s + (0.923 + 0.382i)23-s + (0.923 − 0.382i)25-s + (−0.195 − 0.980i)27-s + (0.555 + 0.831i)29-s − i·31-s − i·33-s + (0.195 + 0.980i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.336 - 0.941i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ 0.336 - 0.941i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.396941545 - 0.9838372387i\)
\(L(\frac12)\) \(\approx\) \(1.396941545 - 0.9838372387i\)
\(L(1)\) \(\approx\) \(1.214741705 - 0.3592525571i\)
\(L(1)\) \(\approx\) \(1.214741705 - 0.3592525571i\)

Euler product

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

−22.01559088269201653886440164935, −21.207875661756636805092475209454, −20.39867985452363577036861404150, −19.743159593735899313269434643362, −19.32278344244609038779820236795, −18.24986494209234412101705859431, −17.25646257462511684885483266130, −16.32878689368636136554042675416, −15.456130048234607070339331816871, −15.1858669859948871855219977226, −14.25186664711457886774060339808, −13.20857053792528383776741673646, −12.60332249475552133580598412427, −11.3830723620347063043581113332, −10.8034304212777447671809742526, −9.763452897746916348006357195823, −8.73217502497889973141292758970, −8.43295489333135414817436143860, −7.305509265867392963801855824631, −6.524800193699761441138059765390, −4.94088506300799016613791389948, −4.303862957494703101710686377039, −3.54736156521482572729197228648, −2.545910514363416454607235675213, −1.26763488631690924616626086524, 0.77775662893956566520702515966, 1.91132585906042034768676340947, 3.353600953264612679840607468540, 3.55195290860157368844879512195, 4.813274847695797630887385929824, 6.316433518402721825951122656995, 6.867384186412400793377649062929, 7.90662490095737916727575862866, 8.556287428711945817390564175385, 9.17924314283796118557273366297, 10.47112716341778512128548332169, 11.49063388917347108638116016515, 11.955941687185791594633768158760, 13.147114766331021424839084528454, 13.69755572147041323198996853448, 14.61512802657780058383097536659, 15.29424711717561135365429617258, 16.110078229191069815641470489121, 16.93222489420878559852245758795, 18.27447174849255958568871633443, 18.689463756930244093324680540121, 19.37344966917512770891546933065, 20.16698853412985150828615439404, 20.7537069349607428243729743164, 21.74518929238983974161937480477

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