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Properties

Label 1-896-896.291-r1-0-0
Degree $1$
Conductor $896$
Sign $0.148 - 0.988i$
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  + (0.321 + 0.946i)3-s + (−0.442 − 0.896i)5-s + (−0.793 + 0.608i)9-s + (−0.751 + 0.659i)11-s + (0.555 + 0.831i)13-s + (0.707 − 0.707i)15-s + (−0.258 + 0.965i)17-s + (−0.0654 − 0.997i)19-s + (−0.608 − 0.793i)23-s + (−0.608 + 0.793i)25-s + (−0.831 − 0.555i)27-s + (−0.195 + 0.980i)29-s + (0.866 − 0.5i)31-s + (−0.866 − 0.5i)33-s + (0.896 − 0.442i)37-s + ⋯
L(s)  = 1  + (0.321 + 0.946i)3-s + (−0.442 − 0.896i)5-s + (−0.793 + 0.608i)9-s + (−0.751 + 0.659i)11-s + (0.555 + 0.831i)13-s + (0.707 − 0.707i)15-s + (−0.258 + 0.965i)17-s + (−0.0654 − 0.997i)19-s + (−0.608 − 0.793i)23-s + (−0.608 + 0.793i)25-s + (−0.831 − 0.555i)27-s + (−0.195 + 0.980i)29-s + (0.866 − 0.5i)31-s + (−0.866 − 0.5i)33-s + (0.896 − 0.442i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.148 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.148 - 0.988i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ 0.148 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4642526480 - 0.3996279206i\)
\(L(\frac12)\) \(\approx\) \(0.4642526480 - 0.3996279206i\)
\(L(1)\) \(\approx\) \(0.8653243434 + 0.1906837124i\)
\(L(1)\) \(\approx\) \(0.8653243434 + 0.1906837124i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.321 + 0.946i)T \)
5 \( 1 + (-0.442 - 0.896i)T \)
11 \( 1 + (-0.751 + 0.659i)T \)
13 \( 1 + (0.555 + 0.831i)T \)
17 \( 1 + (-0.258 + 0.965i)T \)
19 \( 1 + (-0.0654 - 0.997i)T \)
23 \( 1 + (-0.608 - 0.793i)T \)
29 \( 1 + (-0.195 + 0.980i)T \)
31 \( 1 + (0.866 - 0.5i)T \)
37 \( 1 + (0.896 - 0.442i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
43 \( 1 + (-0.980 + 0.195i)T \)
47 \( 1 + (0.965 - 0.258i)T \)
53 \( 1 + (-0.751 + 0.659i)T \)
59 \( 1 + (0.997 + 0.0654i)T \)
61 \( 1 + (0.659 - 0.751i)T \)
67 \( 1 + (-0.321 - 0.946i)T \)
71 \( 1 + (-0.923 + 0.382i)T \)
73 \( 1 + (0.130 + 0.991i)T \)
79 \( 1 + (-0.258 - 0.965i)T \)
83 \( 1 + (0.831 - 0.555i)T \)
89 \( 1 + (-0.991 - 0.130i)T \)
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

−22.15096472394267741454576282712, −20.92093288776672605690768675383, −20.33453520605692866459688661357, −19.314130087501919866061296489498, −18.79458600448217705015831109374, −18.13123789022175972349668273696, −17.529605486986996489364254877355, −16.199977941705662041291268742913, −15.48843755170549894657007811784, −14.66725365661630739917891821962, −13.68638693333327775535908203258, −13.37599177549095662660050091404, −12.12746353522174395895624470315, −11.539496401953540963071030452403, −10.64112764099353818377587160677, −9.7503188157093706920474705986, −8.360759174624320890461687550730, −7.96073332326341378263853010715, −7.12315918030461390560055419939, −6.18513155668380115515105434898, −5.47452645175829990588824092300, −3.84028891743675323892990009321, −3.04739609732475752175086693670, −2.30037451440597416129247587980, −0.91087728776721541230150068644, 0.14605459086449077369053605327, 1.70080712701899460797257797443, 2.80381865112223298800768226649, 4.07815489534075669790791972804, 4.497380902337340175899359866676, 5.383763538355391141634280676157, 6.53649041237377963381655070902, 7.834086025207450997075370213566, 8.5187480602305446136504790783, 9.20149129566170979657165350147, 10.08619005489557020723777074785, 10.96183117108551564440139411467, 11.74746287711416720911339428010, 12.8159001936440506495816208876, 13.456295193801362488050838505570, 14.55890364437349298271291184108, 15.33097148195843996197476845750, 15.95034156015290107131538430772, 16.62937061765652196939540603793, 17.38776448874557957032219302614, 18.463702452218052925598956868285, 19.48432617798939557932966685749, 20.17275103761539624832645612496, 20.69068632209863123528730976509, 21.53209620317939900938868557192

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