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Properties

Label 1-6013-6013.293-r0-0-0
Degree $1$
Conductor $6013$
Sign $0.481 + 0.876i$
Analytic cond. $27.9242$
Root an. cond. $27.9242$
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  + (−0.221 + 0.975i)2-s + (−0.992 + 0.124i)3-s + (−0.901 − 0.431i)4-s + (0.676 − 0.736i)5-s + (0.0987 − 0.995i)6-s + (0.621 − 0.783i)8-s + (0.969 − 0.246i)9-s + (0.568 + 0.822i)10-s + (−0.999 − 0.0439i)11-s + (0.948 + 0.316i)12-s + (0.142 + 0.989i)13-s + (−0.580 + 0.814i)15-s + (0.626 + 0.779i)16-s + (0.925 − 0.378i)17-s + (0.0256 + 0.999i)18-s + (0.0475 + 0.998i)19-s + ⋯
L(s)  = 1  + (−0.221 + 0.975i)2-s + (−0.992 + 0.124i)3-s + (−0.901 − 0.431i)4-s + (0.676 − 0.736i)5-s + (0.0987 − 0.995i)6-s + (0.621 − 0.783i)8-s + (0.969 − 0.246i)9-s + (0.568 + 0.822i)10-s + (−0.999 − 0.0439i)11-s + (0.948 + 0.316i)12-s + (0.142 + 0.989i)13-s + (−0.580 + 0.814i)15-s + (0.626 + 0.779i)16-s + (0.925 − 0.378i)17-s + (0.0256 + 0.999i)18-s + (0.0475 + 0.998i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6013\)    =    \(7 \cdot 859\)
Sign: $0.481 + 0.876i$
Analytic conductor: \(27.9242\)
Root analytic conductor: \(27.9242\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6013} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6013,\ (0:\ ),\ 0.481 + 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9830062442 + 0.5817463593i\)
\(L(\frac12)\) \(\approx\) \(0.9830062442 + 0.5817463593i\)
\(L(1)\) \(\approx\) \(0.7108434236 + 0.2820380438i\)
\(L(1)\) \(\approx\) \(0.7108434236 + 0.2820380438i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
859 \( 1 \)
good2 \( 1 + (-0.221 + 0.975i)T \)
3 \( 1 + (-0.992 + 0.124i)T \)
5 \( 1 + (0.676 - 0.736i)T \)
11 \( 1 + (-0.999 - 0.0439i)T \)
13 \( 1 + (0.142 + 0.989i)T \)
17 \( 1 + (0.925 - 0.378i)T \)
19 \( 1 + (0.0475 + 0.998i)T \)
23 \( 1 + (-0.428 + 0.903i)T \)
29 \( 1 + (0.697 - 0.716i)T \)
31 \( 1 + (0.597 - 0.801i)T \)
37 \( 1 + (0.974 + 0.225i)T \)
41 \( 1 + (0.113 - 0.993i)T \)
43 \( 1 + (0.928 + 0.371i)T \)
47 \( 1 + (-0.993 + 0.116i)T \)
53 \( 1 + (0.660 - 0.750i)T \)
59 \( 1 + (0.512 + 0.858i)T \)
61 \( 1 + (0.959 + 0.281i)T \)
67 \( 1 + (0.448 - 0.893i)T \)
71 \( 1 + (0.996 - 0.0877i)T \)
73 \( 1 + (-0.758 + 0.652i)T \)
79 \( 1 + (0.603 + 0.797i)T \)
83 \( 1 + (0.192 + 0.981i)T \)
89 \( 1 + (0.320 + 0.947i)T \)
97 \( 1 + (0.649 - 0.760i)T \)
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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

−17.67039741151528131419718519418, −17.41002726704588862613526566643, −16.40263488233915504804941166556, −15.84292304810198187849519819037, −14.85034653699476333681068629975, −14.21313889547679244346606229809, −13.26082828256677253657450532656, −12.97019014595202172175973952256, −12.324143865036227962352190812747, −11.52887344309191353767625202983, −10.85290765254071259700729189747, −10.28447081786645270275974258129, −10.16692438632874690985408643599, −9.20347906895784212297242973805, −8.16859091505701230984167092688, −7.64850122880425263994751363074, −6.73471214925944656881295666252, −5.98718050955956655505794143160, −5.22388452264012964264380354938, −4.81155090364323708066714988919, −3.7131003351698353620698579760, −2.834339921132924786359819925040, −2.40459817571971705463600656290, −1.319593366259074544122337934672, −0.61052739454542246021928758230, 0.67635214140463955803699408581, 1.35502415812790315263551636796, 2.32825660305352644396875828948, 3.85084549396645982836329202568, 4.38409475877401578164823510248, 5.25652398776491995441856123653, 5.59635220429725571505017650663, 6.1900997005992785057940615125, 6.91628060513013420483431886493, 7.84429664786292298170413886725, 8.21983113383773068317843883391, 9.360082100775640421972317976746, 9.81057847968986778259067226440, 10.185830780257662969500091529516, 11.1933084101773978859919205751, 12.01673069550007519687296753114, 12.60363647072637904726142406193, 13.401712593280514832132068084450, 13.809868555535155261620640313013, 14.61333246267349296006324485654, 15.53028358788797359961299280392, 16.09300390982867590963250768156, 16.53106443550389943511169968167, 16.9864172789472593801597179633, 17.77422381379173786364130627503

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