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Properties

Label 1-1045-1045.1003-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.984 - 0.175i$
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  + (−0.970 + 0.241i)2-s + (0.999 − 0.0348i)3-s + (0.882 − 0.469i)4-s + (−0.961 + 0.275i)6-s + (0.743 + 0.669i)7-s + (−0.743 + 0.669i)8-s + (0.997 − 0.0697i)9-s + (0.866 − 0.5i)12-s + (−0.829 + 0.559i)13-s + (−0.882 − 0.469i)14-s + (0.559 − 0.829i)16-s + (0.0697 − 0.997i)17-s + (−0.951 + 0.309i)18-s + (0.766 + 0.642i)21-s + (−0.984 + 0.173i)23-s + (−0.719 + 0.694i)24-s + ⋯
L(s)  = 1  + (−0.970 + 0.241i)2-s + (0.999 − 0.0348i)3-s + (0.882 − 0.469i)4-s + (−0.961 + 0.275i)6-s + (0.743 + 0.669i)7-s + (−0.743 + 0.669i)8-s + (0.997 − 0.0697i)9-s + (0.866 − 0.5i)12-s + (−0.829 + 0.559i)13-s + (−0.882 − 0.469i)14-s + (0.559 − 0.829i)16-s + (0.0697 − 0.997i)17-s + (−0.951 + 0.309i)18-s + (0.766 + 0.642i)21-s + (−0.984 + 0.173i)23-s + (−0.719 + 0.694i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.984 - 0.175i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (1003, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.984 - 0.175i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02212621659 + 0.2497897108i\)
\(L(\frac12)\) \(\approx\) \(0.02212621659 + 0.2497897108i\)
\(L(1)\) \(\approx\) \(0.8562457134 + 0.1443641072i\)
\(L(1)\) \(\approx\) \(0.8562457134 + 0.1443641072i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.970 + 0.241i)T \)
3 \( 1 + (0.999 - 0.0348i)T \)
7 \( 1 + (0.743 + 0.669i)T \)
13 \( 1 + (-0.829 + 0.559i)T \)
17 \( 1 + (0.0697 - 0.997i)T \)
23 \( 1 + (-0.984 + 0.173i)T \)
29 \( 1 + (-0.848 - 0.529i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (0.0348 + 0.999i)T \)
43 \( 1 + (-0.984 - 0.173i)T \)
47 \( 1 + (-0.927 + 0.374i)T \)
53 \( 1 + (-0.898 - 0.438i)T \)
59 \( 1 + (-0.374 + 0.927i)T \)
61 \( 1 + (0.719 + 0.694i)T \)
67 \( 1 + (0.642 + 0.766i)T \)
71 \( 1 + (-0.438 - 0.898i)T \)
73 \( 1 + (-0.788 + 0.615i)T \)
79 \( 1 + (-0.961 - 0.275i)T \)
83 \( 1 + (0.406 - 0.913i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (0.970 - 0.241i)T \)
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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

−20.60160382216179414687851389125, −20.10044758334684994558741151915, −19.504315410889698412752642519310, −18.71468343915845138062077444000, −17.81520676762086039238088222087, −17.25326490610409799286857391040, −16.325192235408681349842399135197, −15.43896489636438888733483810357, −14.69820135684122301354568367145, −14.013415786815800602295574389202, −12.851325840938312276444962029429, −12.246734369567978659321837447610, −11.04439701313690540564024695961, −10.32027389531719751466247956435, −9.762615099394712915956201057628, −8.66074493195738591561188820410, −8.11379591932979388885704499998, −7.43216161795272529606079289032, −6.65109226226818483092441666917, −5.17884882257916051086520588620, −3.95896972634512287390699241324, −3.23013515812760900005078527669, −2.03280644848322095892626762229, −1.465250768481875007676760526591, −0.05431540881438079387640240477, 1.496667703624857275851036545897, 2.1850248553089231770181427136, 2.99288089237973475476720880443, 4.40864331617657008097971363914, 5.39974016329301704725825722539, 6.554677192952053771399924164047, 7.505114167383645703828497561601, 8.01376022149321408463063723532, 8.88358630819936847421266878982, 9.560012681431887193579357378348, 10.163740032143761340358043148722, 11.540967826954041518129915363676, 11.87700721074343709673783061167, 13.173067184413086570445257473451, 14.27047612655432322874410980958, 14.71236318461333259236596373747, 15.50577247493211450790822562801, 16.21547199291908717322585954946, 17.18239866461431148469502113590, 18.07089062721304358467347422668, 18.65550360131115696062282816778, 19.28553106592110712195560851795, 20.143629276893090739407314099300, 20.75360794905549344997628094791, 21.43441377697715549177270093488

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