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Properties

Label 1-6012-6012.707-r1-0-0
Degree $1$
Conductor $6012$
Sign $0.995 + 0.0921i$
Analytic cond. $646.079$
Root an. cond. $646.079$
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  + (−0.605 + 0.796i)5-s + (0.682 − 0.731i)7-s + (−0.924 − 0.381i)11-s + (0.814 − 0.579i)13-s + (−0.800 + 0.599i)17-s + (−0.489 + 0.872i)19-s + (0.564 + 0.825i)23-s + (−0.267 − 0.963i)25-s + (0.564 − 0.825i)29-s + (0.986 − 0.163i)31-s + (0.169 + 0.985i)35-s + (−0.776 + 0.629i)37-s + (−0.194 + 0.980i)41-s + (0.875 + 0.483i)43-s + (0.994 − 0.100i)47-s + ⋯
L(s)  = 1  + (−0.605 + 0.796i)5-s + (0.682 − 0.731i)7-s + (−0.924 − 0.381i)11-s + (0.814 − 0.579i)13-s + (−0.800 + 0.599i)17-s + (−0.489 + 0.872i)19-s + (0.564 + 0.825i)23-s + (−0.267 − 0.963i)25-s + (0.564 − 0.825i)29-s + (0.986 − 0.163i)31-s + (0.169 + 0.985i)35-s + (−0.776 + 0.629i)37-s + (−0.194 + 0.980i)41-s + (0.875 + 0.483i)43-s + (0.994 − 0.100i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6012 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6012\)    =    \(2^{2} \cdot 3^{2} \cdot 167\)
Sign: $0.995 + 0.0921i$
Analytic conductor: \(646.079\)
Root analytic conductor: \(646.079\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6012} (707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6012,\ (1:\ ),\ 0.995 + 0.0921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.844787478 + 0.08514353336i\)
\(L(\frac12)\) \(\approx\) \(1.844787478 + 0.08514353336i\)
\(L(1)\) \(\approx\) \(0.9841559092 + 0.04448486266i\)
\(L(1)\) \(\approx\) \(0.9841559092 + 0.04448486266i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (-0.605 + 0.796i)T \)
7 \( 1 + (0.682 - 0.731i)T \)
11 \( 1 + (-0.924 - 0.381i)T \)
13 \( 1 + (0.814 - 0.579i)T \)
17 \( 1 + (-0.800 + 0.599i)T \)
19 \( 1 + (-0.489 + 0.872i)T \)
23 \( 1 + (0.564 + 0.825i)T \)
29 \( 1 + (0.564 - 0.825i)T \)
31 \( 1 + (0.986 - 0.163i)T \)
37 \( 1 + (-0.776 + 0.629i)T \)
41 \( 1 + (-0.194 + 0.980i)T \)
43 \( 1 + (0.875 + 0.483i)T \)
47 \( 1 + (0.994 - 0.100i)T \)
53 \( 1 + (0.614 - 0.788i)T \)
59 \( 1 + (0.119 - 0.992i)T \)
61 \( 1 + (-0.543 - 0.839i)T \)
67 \( 1 + (0.992 - 0.125i)T \)
71 \( 1 + (-0.822 + 0.569i)T \)
73 \( 1 + (-0.726 - 0.686i)T \)
79 \( 1 + (0.984 - 0.175i)T \)
83 \( 1 + (-0.653 + 0.756i)T \)
89 \( 1 + (-0.0567 + 0.998i)T \)
97 \( 1 + (-0.986 - 0.163i)T \)
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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

−17.65634360795734574675379895296, −16.86551617600818822476518505844, −16.04257487696988274676231565077, −15.54653106240380592208536837239, −15.25877802777476903900674522398, −14.200828901848951945851302598, −13.57489636425866454004184582230, −12.85796180518501372942735831585, −12.232176091626649954361253958131, −11.716166137468917358093067453018, −10.87953774277157670665768393555, −10.5227070282178781988496225669, −9.17972192839886404159475512472, −8.75802029673706358098784112238, −8.47538044836383949272497024636, −7.41266338442213820644128538318, −6.93770861148457586874217560843, −5.87465888058732618097597288420, −5.15659278429822599962328985767, −4.57801554354938347088615812736, −4.09183300003004621609370084162, −2.81387727124439277913461931335, −2.30600910829804112688372188114, −1.3173082584871629935136638442, −0.48322658939760977279856684964, 0.4661850332646852639052130534, 1.33619198555914384387694954965, 2.32470053408489057484023850862, 3.12603701795109138008501327341, 3.82778632822226263728215430505, 4.42589212320397284202413239725, 5.31800250380483789468121116496, 6.17721057080094565428196090113, 6.74299389260447601972120925373, 7.66914197506762813704222484250, 8.14180848914518365041359367247, 8.508219591377624105447103517851, 9.84691090091701606270372829701, 10.437631663127997085134195795217, 10.958261399827776094224161598132, 11.36453166701509961484526972832, 12.233810760621177821352642482547, 13.132898087290174796326134408474, 13.63964725403252597375966164935, 14.23152676976932566765208626804, 15.20013339031366751861126761810, 15.384144058954777090526898413147, 16.1304983151551303448548809820, 16.99818605338520502831834696864, 17.68772999819160722309354104931

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