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Properties

Label 1-1045-1045.117-r1-0-0
Degree $1$
Conductor $1045$
Sign $0.0737 + 0.997i$
Analytic cond. $112.300$
Root an. cond. $112.300$
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  + (−0.469 − 0.882i)2-s + (−0.0697 − 0.997i)3-s + (−0.559 + 0.829i)4-s + (−0.848 + 0.529i)6-s + (−0.994 + 0.104i)7-s + (0.994 + 0.104i)8-s + (−0.990 + 0.139i)9-s + (0.866 + 0.5i)12-s + (0.927 + 0.374i)13-s + (0.559 + 0.829i)14-s + (−0.374 − 0.927i)16-s + (0.139 − 0.990i)17-s + (0.587 + 0.809i)18-s + (0.173 + 0.984i)21-s + (0.342 + 0.939i)23-s + (0.0348 − 0.999i)24-s + ⋯
L(s)  = 1  + (−0.469 − 0.882i)2-s + (−0.0697 − 0.997i)3-s + (−0.559 + 0.829i)4-s + (−0.848 + 0.529i)6-s + (−0.994 + 0.104i)7-s + (0.994 + 0.104i)8-s + (−0.990 + 0.139i)9-s + (0.866 + 0.5i)12-s + (0.927 + 0.374i)13-s + (0.559 + 0.829i)14-s + (−0.374 − 0.927i)16-s + (0.139 − 0.990i)17-s + (0.587 + 0.809i)18-s + (0.173 + 0.984i)21-s + (0.342 + 0.939i)23-s + (0.0348 − 0.999i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0737 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0737 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.0737 + 0.997i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ 0.0737 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1508528542 - 0.1401097341i\)
\(L(\frac12)\) \(\approx\) \(-0.1508528542 - 0.1401097341i\)
\(L(1)\) \(\approx\) \(0.4883799130 - 0.4130140363i\)
\(L(1)\) \(\approx\) \(0.4883799130 - 0.4130140363i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.469 - 0.882i)T \)
3 \( 1 + (-0.0697 - 0.997i)T \)
7 \( 1 + (-0.994 + 0.104i)T \)
13 \( 1 + (0.927 + 0.374i)T \)
17 \( 1 + (0.139 - 0.990i)T \)
23 \( 1 + (0.342 + 0.939i)T \)
29 \( 1 + (-0.438 - 0.898i)T \)
31 \( 1 + (0.978 + 0.207i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.997 + 0.0697i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (-0.694 - 0.719i)T \)
53 \( 1 + (0.788 - 0.615i)T \)
59 \( 1 + (-0.719 - 0.694i)T \)
61 \( 1 + (-0.0348 - 0.999i)T \)
67 \( 1 + (-0.984 - 0.173i)T \)
71 \( 1 + (0.615 - 0.788i)T \)
73 \( 1 + (-0.970 - 0.241i)T \)
79 \( 1 + (-0.848 - 0.529i)T \)
83 \( 1 + (-0.743 + 0.669i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (0.469 + 0.882i)T \)
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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

−22.126244706944947978074264324450, −21.23388104262040828640758526967, −20.241579217918718959903260991452, −19.590796415054405560037212496583, −18.74844143334843174393898103807, −17.88689897496992121993520158823, −16.956230145116636032101159498476, −16.43709613803212542594046369810, −15.7962218410656750391565958465, −15.09737099459794905854557598213, −14.403011776532313523427271294637, −13.4092759587562632984996653778, −12.64605685551309119510584125122, −11.219313122266808686924175469708, −10.41923555771783026710825754020, −9.957234009593062188171905511016, −8.91013136546944647955020353179, −8.50307431374841167248857600911, −7.32414474319632683592368282861, −6.15279928917036330930782736502, −5.89781894489341074297113986425, −4.65032617019720320904885991988, −3.868629778848620261461974287572, −2.86942139730883477239327173512, −1.130012701620744616434834930, 0.06629444319043562155808864768, 0.971302174779427704187097779453, 1.98233216518282405548840008463, 2.96784773837722301382473040243, 3.639239287749623093336565481175, 5.02340145161798262761584888955, 6.17946750911827651375582773385, 6.980077267181793248705341071472, 7.85032654022484956366289149976, 8.745619998811499997585990299772, 9.459203705750538208105033091380, 10.328751689550205042564349645765, 11.52351173675703405464814219183, 11.7761586945739105050620175577, 12.80450193768417150307428709135, 13.52251872523255350104691461046, 13.8236471357078981723754685962, 15.36888648998082471976843896400, 16.36546860555284701641289497781, 17.03562162134221417282875216712, 17.89409124792651828927213022174, 18.72059199044147232346420138436, 18.988683091401815889094414060132, 19.87372770980273625383900873221, 20.5233038365622884101940638395

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