Properties

Label 1-1045-1045.269-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.943 - 0.332i$
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  + (−0.719 − 0.694i)2-s + (0.848 + 0.529i)3-s + (0.0348 + 0.999i)4-s + (−0.241 − 0.970i)6-s + (−0.669 − 0.743i)7-s + (0.669 − 0.743i)8-s + (0.438 + 0.898i)9-s + (−0.5 + 0.866i)12-s + (−0.997 − 0.0697i)13-s + (−0.0348 + 0.999i)14-s + (−0.997 + 0.0697i)16-s + (−0.438 + 0.898i)17-s + (0.309 − 0.951i)18-s + (−0.173 − 0.984i)21-s + (0.939 − 0.342i)23-s + (0.961 − 0.275i)24-s + ⋯
L(s)  = 1  + (−0.719 − 0.694i)2-s + (0.848 + 0.529i)3-s + (0.0348 + 0.999i)4-s + (−0.241 − 0.970i)6-s + (−0.669 − 0.743i)7-s + (0.669 − 0.743i)8-s + (0.438 + 0.898i)9-s + (−0.5 + 0.866i)12-s + (−0.997 − 0.0697i)13-s + (−0.0348 + 0.999i)14-s + (−0.997 + 0.0697i)16-s + (−0.438 + 0.898i)17-s + (0.309 − 0.951i)18-s + (−0.173 − 0.984i)21-s + (0.939 − 0.342i)23-s + (0.961 − 0.275i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.943 - 0.332i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.943 - 0.332i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05601798989 - 0.3274509420i\)
\(L(\frac12)\) \(\approx\) \(0.05601798989 - 0.3274509420i\)
\(L(1)\) \(\approx\) \(0.7799547322 - 0.09797959242i\)
\(L(1)\) \(\approx\) \(0.7799547322 - 0.09797959242i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.719 - 0.694i)T \)
3 \( 1 + (0.848 + 0.529i)T \)
7 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (-0.997 - 0.0697i)T \)
17 \( 1 + (-0.438 + 0.898i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (0.882 + 0.469i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (-0.848 - 0.529i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (-0.990 + 0.139i)T \)
53 \( 1 + (0.559 - 0.829i)T \)
59 \( 1 + (-0.990 - 0.139i)T \)
61 \( 1 + (0.961 + 0.275i)T \)
67 \( 1 + (0.173 - 0.984i)T \)
71 \( 1 + (-0.559 - 0.829i)T \)
73 \( 1 + (0.374 + 0.927i)T \)
79 \( 1 + (0.241 - 0.970i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (-0.719 - 0.694i)T \)
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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

−21.697545687329870989528534937861, −20.589302877673146272878603074147, −19.874058430882964570598192515873, −19.149875880422541388390499775446, −18.71988794151638188074667429761, −17.893229713068195123892106968560, −17.100135383114759623365977185253, −16.14698594085350174619701599281, −15.277943269131269778796692634119, −14.93534505450071374352505105641, −13.87732094341542585630952535635, −13.24300783101187196158276406097, −12.211999601342568244486818483165, −11.38252246686778947634652364569, −9.92704357911320430478809747491, −9.538406842446430078234209787645, −8.76104154735707551632670315908, −7.98027371292710101852168336269, −7.07050723828981073490337543219, −6.55044816453685014855815997989, −5.48756480910565159536439798715, −4.45299251652268186706315130701, −2.88921739194914557614933766248, −2.338928502114412740792657948609, −1.056272811798418726329719430060, 0.083366780221276245044319897593, 1.40936650008209655905561899140, 2.522957009801303194656858699529, 3.24828391170130159009051392872, 4.0902266877361374111331564713, 4.92215797734789892148419780253, 6.69040703785359531534918576873, 7.35592303781597139612229277290, 8.293231351492893713092677966393, 9.02281383139680799398560460547, 9.81471036876191079503245415937, 10.43432951494446365117418863992, 11.0350627602054031144129480575, 12.42256255988198782549982222604, 12.91894652812996105103771911675, 13.82199962799797202709298643658, 14.67405406288309615468967462914, 15.67462909432597910887386935514, 16.42229915355910903198771005942, 17.09510554171146128880267353146, 17.9077773643412214698728518171, 19.097228441967129860887773439641, 19.57335712273685273610888645263, 19.96701007354767559769019129458, 20.90067302277786123698419288683

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