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Properties

Label 1-896-896.739-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: $\chi_{896} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (1:\ ),\ -0.148 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5613774824 + 0.6521590942i\)
\(L(\frac12)\) \(\approx\) \(0.5613774824 + 0.6521590942i\)
\(L(1)\) \(\approx\) \(0.9023927702 - 0.07318765183i\)
\(L(1)\) \(\approx\) \(0.9023927702 - 0.07318765183i\)

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

−21.60570042113334113358663222211, −20.81438917188542763561318296734, −20.05277983903172703422373035386, −19.5221301588073422010485359655, −18.03638411773967713027079105187, −17.365320909912964499524259409990, −16.85204664865989342845024467963, −15.91857847484013160971483494556, −15.427229696374613718568154509424, −14.16300475201291138958587965834, −13.79178528256674013974152160455, −12.28847677619259445249173872571, −11.991557992819552634298799347454, −10.93269865038061146827718543101, −9.92612541187208180416239855895, −9.15908559654495310121772748687, −8.92554271392902706769369398020, −7.35781913472118210965729878, −6.45336844332465464556081947657, −5.33955402255474643290962244733, −4.708889053101566204899723877902, −4.02899038629274155280890070841, −2.6876162958714832474692830105, −1.49267996270921796416476946300, −0.20388982601179608058541760746, 1.088804132352216527249450719806, 2.155174980909916449386498683884, 2.988887636617807624992148321139, 4.1498741815259464542673854129, 5.71760297933884219123202577026, 6.09142184242592763929929486595, 6.929562503020125902172706763784, 7.898114611242419969596447039020, 8.57820776247409931899623109861, 10.00048769647494770982252003201, 10.55465221317880089269087808677, 11.56155738036358796797999183945, 12.225116914310967679940655194999, 13.16060345051221832518006570651, 13.95873517837144733998097060784, 14.55173141122111409804139844489, 15.452173014810381144060347238276, 16.78705842651654354020311345311, 17.27087382793420502399500130707, 18.02119849114975865470749868017, 18.87397969231212466323345824277, 19.305545970477262265146556669803, 20.22960646636895255422362664995, 21.33800113402261171733447007342, 22.31500772033707503275772611786

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