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Properties

Label 1-1045-1045.978-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.328 - 0.944i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
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  + (−0.342 − 0.939i)2-s + (0.984 + 0.173i)3-s + (−0.766 + 0.642i)4-s + (−0.173 − 0.984i)6-s + (0.866 − 0.5i)7-s + (0.866 + 0.5i)8-s + (0.939 + 0.342i)9-s + (−0.866 + 0.5i)12-s + (0.984 − 0.173i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (−0.342 − 0.939i)17-s − i·18-s + (0.939 − 0.342i)21-s + (0.642 + 0.766i)23-s + (0.766 + 0.642i)24-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (0.984 + 0.173i)3-s + (−0.766 + 0.642i)4-s + (−0.173 − 0.984i)6-s + (0.866 − 0.5i)7-s + (0.866 + 0.5i)8-s + (0.939 + 0.342i)9-s + (−0.866 + 0.5i)12-s + (0.984 − 0.173i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (−0.342 − 0.939i)17-s − i·18-s + (0.939 − 0.342i)21-s + (0.642 + 0.766i)23-s + (0.766 + 0.642i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.328 - 0.944i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (978, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.328 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.648098224 - 1.171499593i\)
\(L(\frac12)\) \(\approx\) \(1.648098224 - 1.171499593i\)
\(L(1)\) \(\approx\) \(1.252652047 - 0.5620247600i\)
\(L(1)\) \(\approx\) \(1.252652047 - 0.5620247600i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.342 - 0.939i)T \)
3 \( 1 + (0.984 + 0.173i)T \)
7 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (0.984 - 0.173i)T \)
17 \( 1 + (-0.342 - 0.939i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (-0.939 - 0.342i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (-0.342 + 0.939i)T \)
53 \( 1 + (0.642 + 0.766i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (-0.766 + 0.642i)T \)
67 \( 1 + (-0.342 + 0.939i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 + (-0.984 - 0.173i)T \)
79 \( 1 + (0.173 - 0.984i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (0.342 + 0.939i)T \)
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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

−21.59040690881448824450331043386, −20.90609689902464301683287402063, −20.024967809187294137685895242843, −19.15045002980772540800019412318, −18.458038799398691306333739708318, −17.98797991384976209885086681167, −16.992834535970411182452100063275, −16.08778918130438548468436202764, −15.216068648777904089969235382900, −14.81248552598750923582584603389, −14.03988959433861486374526332644, −13.26326953992861162306908599433, −12.49595245860066498292219598031, −11.112761506786656957355788928329, −10.34564489454455125169427496334, −9.18510744551926550236846543472, −8.618694933021664594014080615081, −8.15440051451617679215866717727, −7.15920439111691204841998001576, −6.37458471587892230645131426634, −5.3366601445284215012672347979, −4.379871688316411200046853441994, −3.48465845707512708456566543618, −2.02874036867072474446245398504, −1.26880655695204122707736330798, 1.0246757400639495723463933546, 1.89979959182005734176720989251, 2.84980295301480082594722169491, 3.813977523793132925207980922493, 4.411507628086861124124476681641, 5.49443607372599580230112623605, 7.27570051686813060864404251401, 7.73187736055460697021870478105, 8.721702291356943559420112487366, 9.25070286145846619623230024751, 10.16668549648462873696643644813, 11.067372193888107811721189647897, 11.52121309819443569735293429047, 12.88297616773564795815673181178, 13.43231605457967599162600274060, 14.08451641074421581258319090921, 14.92248171557767143665954618361, 15.91453000350777983735953188816, 16.829741790881936920689960508506, 17.80746633065809828372393308243, 18.40652205268187673128567285626, 19.1609168150771971010043546788, 20.02085255919542386279862768670, 20.651829956011993781387592851039, 20.98056284147491737301316518769

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