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Properties

Label 1-3021-3021.383-r1-0-0
Degree $1$
Conductor $3021$
Sign $0.822 - 0.568i$
Analytic cond. $324.651$
Root an. cond. $324.651$
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  + (−0.852 − 0.523i)2-s + (0.452 + 0.891i)4-s + (0.133 + 0.991i)5-s + (−0.948 + 0.316i)7-s + (0.0804 − 0.996i)8-s + (0.404 − 0.914i)10-s + (−0.632 + 0.774i)11-s + (−0.730 + 0.682i)13-s + (0.974 + 0.226i)14-s + (−0.589 + 0.807i)16-s + (−0.711 + 0.702i)17-s + (−0.822 + 0.568i)20-s + (0.944 − 0.329i)22-s + (0.342 + 0.939i)23-s + (−0.964 + 0.265i)25-s + (0.979 − 0.200i)26-s + ⋯
L(s)  = 1  + (−0.852 − 0.523i)2-s + (0.452 + 0.891i)4-s + (0.133 + 0.991i)5-s + (−0.948 + 0.316i)7-s + (0.0804 − 0.996i)8-s + (0.404 − 0.914i)10-s + (−0.632 + 0.774i)11-s + (−0.730 + 0.682i)13-s + (0.974 + 0.226i)14-s + (−0.589 + 0.807i)16-s + (−0.711 + 0.702i)17-s + (−0.822 + 0.568i)20-s + (0.944 − 0.329i)22-s + (0.342 + 0.939i)23-s + (−0.964 + 0.265i)25-s + (0.979 − 0.200i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3021\)    =    \(3 \cdot 19 \cdot 53\)
Sign: $0.822 - 0.568i$
Analytic conductor: \(324.651\)
Root analytic conductor: \(324.651\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3021} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3021,\ (1:\ ),\ 0.822 - 0.568i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06135181626 + 0.01915512754i\)
\(L(\frac12)\) \(\approx\) \(-0.06135181626 + 0.01915512754i\)
\(L(1)\) \(\approx\) \(0.4450353801 + 0.1276787737i\)
\(L(1)\) \(\approx\) \(0.4450353801 + 0.1276787737i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 \)
53 \( 1 \)
good2 \( 1 + (-0.852 - 0.523i)T \)
5 \( 1 + (0.133 + 0.991i)T \)
7 \( 1 + (-0.948 + 0.316i)T \)
11 \( 1 + (-0.632 + 0.774i)T \)
13 \( 1 + (-0.730 + 0.682i)T \)
17 \( 1 + (-0.711 + 0.702i)T \)
23 \( 1 + (0.342 + 0.939i)T \)
29 \( 1 + (-0.859 - 0.511i)T \)
31 \( 1 + (0.774 - 0.632i)T \)
37 \( 1 + (-0.970 + 0.239i)T \)
41 \( 1 + (-0.488 + 0.872i)T \)
43 \( 1 + (-0.830 - 0.556i)T \)
47 \( 1 + (-0.611 + 0.791i)T \)
59 \( 1 + (0.303 - 0.952i)T \)
61 \( 1 + (-0.967 - 0.252i)T \)
67 \( 1 + (0.0536 - 0.998i)T \)
71 \( 1 + (-0.556 + 0.830i)T \)
73 \( 1 + (0.265 - 0.964i)T \)
79 \( 1 + (-0.989 - 0.147i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (-0.0938 + 0.995i)T \)
97 \( 1 + (-0.611 - 0.791i)T \)
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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

−18.1823769279340172100380120820, −17.441176665878033148354636388262, −16.754243846015492657066453420051, −16.29187607188748107878104818335, −15.720952208584210229397404923913, −15.0450437281366057332086571355, −13.968315367141882556167623646804, −13.37733907324937559123996643866, −12.67206023014679838488225252912, −11.864940587756475377989037583, −10.87297642368020736668342039172, −10.19674245400829882152760638024, −9.6513546409032380300689763903, −8.69701101499210834301232174535, −8.48023153296307561650898091498, −7.351806232904945775543425469451, −6.8461559567584529943131149421, −5.848662598648065864318049288734, −5.26903794416827663013181527702, −4.52823585402464247694056689630, −3.19415688345565531557184400652, −2.43992997335015698854804958653, −1.292146294933740642720816738483, −0.28390065603344217168436099745, −0.040679072124736993142264822942, 1.70704494411495390306665974790, 2.25514551107446418317371744886, 3.03654353633241939439804552139, 3.726636021569915505553216075999, 4.75295797429803729350577258474, 6.0317154881618868321882298541, 6.703562937276118297948277756159, 7.27298343894641047332495569486, 8.002313211853159917522050687175, 9.019038264411176678034159160146, 9.82609698481865380002784804769, 9.97866266722714489647993800680, 10.94519000824341532095945589215, 11.5978428885889690418677331671, 12.31268763933431901696969355764, 13.11234993505259398058111258457, 13.63575448451539922319610211819, 14.93405918714028085732819430838, 15.35260998021702324377975879943, 16.002583140885392858403361970581, 17.12752261103230695489544989635, 17.339370830790230718965935638201, 18.34247047427341812604407944383, 18.78941073295208900612091998475, 19.418849640191901529823068373172

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