Properties

Label 1-896-896.459-r1-0-0
Degree $1$
Conductor $896$
Sign $-0.969 - 0.244i$
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  + (−0.659 − 0.751i)3-s + (−0.997 + 0.0654i)5-s + (−0.130 + 0.991i)9-s + (−0.946 + 0.321i)11-s + (−0.555 + 0.831i)13-s + (0.707 + 0.707i)15-s + (0.965 + 0.258i)17-s + (−0.896 + 0.442i)19-s + (0.991 + 0.130i)23-s + (0.991 − 0.130i)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 + (0.0654 + 0.997i)37-s + ⋯
L(s)  = 1  + (−0.659 − 0.751i)3-s + (−0.997 + 0.0654i)5-s + (−0.130 + 0.991i)9-s + (−0.946 + 0.321i)11-s + (−0.555 + 0.831i)13-s + (0.707 + 0.707i)15-s + (0.965 + 0.258i)17-s + (−0.896 + 0.442i)19-s + (0.991 + 0.130i)23-s + (0.991 − 0.130i)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 + (0.0654 + 0.997i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.969 - 0.244i$
Analytic conductor: \(96.2885\)
Root analytic conductor: \(96.2885\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ -0.969 - 0.244i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01507561975 + 0.1211949600i\)
\(L(\frac12)\) \(\approx\) \(0.01507561975 + 0.1211949600i\)
\(L(1)\) \(\approx\) \(0.5919705505 + 0.003877390887i\)
\(L(1)\) \(\approx\) \(0.5919705505 + 0.003877390887i\)

Euler product

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

−21.182932666212855119379737241251, −20.78832650828561052994429276057, −19.76320552937172357141438080241, −18.980967309399806466477906597367, −18.13055582091798982128897155935, −17.16558192518712536800265505269, −16.522530688647412132173161040403, −15.68205704342607457079535844438, −15.18452724061424554218138434326, −14.415581286716777412907771663314, −12.898979794281120182289468978731, −12.47996795039163006690373196005, −11.358043883648290096441164257747, −10.88061773965528269531997173506, −10.044149690834951759659258174455, −9.08330561046124325075854837565, −8.03364270756972824101786745785, −7.362311070110547253327223283954, −6.127315641813660398534057497871, −5.20872628438938914053692599200, −4.55384906588633417037370646062, −3.50390259312383529596513923896, −2.70152456829500446245379212840, −0.7130850832593789112881920425, −0.0465224731774394756542066357, 1.22076505266098760653964578049, 2.3435537865610225353286052473, 3.49730067867875350028966818158, 4.71074044031724389519455072622, 5.351780405430187996452334355012, 6.605846794422847390854847425692, 7.28103555204906514602734652952, 7.94982345022362677851931517914, 8.86113223172688264854417920086, 10.275790021255845933823185462355, 10.88595513919669933291157626870, 11.80039216756745595828794077629, 12.46979191189061451682648024976, 13.00124550273570920827998884268, 14.23772530046102398306683536214, 14.9490223329932836029898268056, 15.98920528196193858777286921266, 16.653210021616619637291930623980, 17.356602942395025643313986833882, 18.48046162087176641217645246825, 18.9101116481543781534943436885, 19.53925625450981175313518515181, 20.55842856465373037976502727206, 21.4717899914464692273621320540, 22.34375945281328143445329360127

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