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Properties

Label 1-1045-1045.348-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.749 + 0.661i$
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  + (0.139 + 0.990i)2-s + (0.970 − 0.241i)3-s + (−0.961 + 0.275i)4-s + (0.374 + 0.927i)6-s + (−0.406 + 0.913i)7-s + (−0.406 − 0.913i)8-s + (0.882 − 0.469i)9-s + (−0.866 + 0.5i)12-s + (0.529 − 0.848i)13-s + (−0.961 − 0.275i)14-s + (0.848 − 0.529i)16-s + (0.469 − 0.882i)17-s + (0.587 + 0.809i)18-s + (−0.173 + 0.984i)21-s + (0.342 − 0.939i)23-s + (−0.615 − 0.788i)24-s + ⋯
L(s)  = 1  + (0.139 + 0.990i)2-s + (0.970 − 0.241i)3-s + (−0.961 + 0.275i)4-s + (0.374 + 0.927i)6-s + (−0.406 + 0.913i)7-s + (−0.406 − 0.913i)8-s + (0.882 − 0.469i)9-s + (−0.866 + 0.5i)12-s + (0.529 − 0.848i)13-s + (−0.961 − 0.275i)14-s + (0.848 − 0.529i)16-s + (0.469 − 0.882i)17-s + (0.587 + 0.809i)18-s + (−0.173 + 0.984i)21-s + (0.342 − 0.939i)23-s + (−0.615 − 0.788i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.749 + 0.661i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (348, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.749 + 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.933017049 + 0.7310113759i\)
\(L(\frac12)\) \(\approx\) \(1.933017049 + 0.7310113759i\)
\(L(1)\) \(\approx\) \(1.347370323 + 0.5186219082i\)
\(L(1)\) \(\approx\) \(1.347370323 + 0.5186219082i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.139 + 0.990i)T \)
3 \( 1 + (0.970 - 0.241i)T \)
7 \( 1 + (-0.406 + 0.913i)T \)
13 \( 1 + (0.529 - 0.848i)T \)
17 \( 1 + (0.469 - 0.882i)T \)
23 \( 1 + (0.342 - 0.939i)T \)
29 \( 1 + (-0.719 - 0.694i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (0.241 + 0.970i)T \)
43 \( 1 + (0.342 + 0.939i)T \)
47 \( 1 + (-0.898 - 0.438i)T \)
53 \( 1 + (0.999 + 0.0348i)T \)
59 \( 1 + (-0.438 - 0.898i)T \)
61 \( 1 + (0.615 - 0.788i)T \)
67 \( 1 + (0.984 - 0.173i)T \)
71 \( 1 + (0.0348 + 0.999i)T \)
73 \( 1 + (-0.0697 + 0.997i)T \)
79 \( 1 + (-0.374 + 0.927i)T \)
83 \( 1 + (0.207 + 0.978i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (-0.139 - 0.990i)T \)
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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

−21.22439885718290722205580416908, −20.743844714995604191718250391679, −19.82223872635580407806160554660, −19.38059969283015889722056604819, −18.757091768226531059988838173745, −17.730787400336333064376388785369, −16.78519792504346605560314881966, −15.93327167618696218628134134530, −14.83708656447188577713769697586, −14.185884453719081158913826283957, −13.4824958689349690056631389877, −12.94432997880101198157150743274, −11.95864980761222014701790169404, −10.81622308659349645116834666775, −10.36634321445518102576819755699, −9.40612590117332276275937158770, −8.86995218516856056915208334890, −7.861010716600515227725258758, −6.95053043932742762087211126238, −5.65146511259446123487463567768, −4.44160717145773576375449869710, −3.777621962797920468148819748495, −3.2055062429767935841026931601, −1.97641615062376477440740926167, −1.17781008854958424116638902819, 0.88547625089807130383150455596, 2.54875472188025892980040796206, 3.20969254796266811310253967810, 4.269564948000631389178484503868, 5.322950230077495220074678442693, 6.220846005176426207074034625762, 6.96845682274615370449094674378, 8.16491898143011703372280483168, 8.28988666132518565686902893498, 9.555926810541793713164316721867, 9.7910834086465135545943977352, 11.455469378381597165556467027017, 12.60683573447531278027211793300, 13.033421679813943035154089822019, 13.85043113147596055053426191108, 14.74096986232663986852592654149, 15.26696552598742578627260143728, 15.9339052727633180092318683209, 16.74113116750041301147499539978, 17.90970455724521708900636902605, 18.548521061362729067051327983656, 18.95580412276831447510938795009, 20.09029815026792933402544211190, 20.91751013304442040495315338434, 21.65602093829479302368671386173

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