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Properties

Label 1-6003-6003.1210-r0-0-0
Degree $1$
Conductor $6003$
Sign $-0.624 - 0.780i$
Analytic cond. $27.8778$
Root an. cond. $27.8778$
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  + (0.585 − 0.810i)2-s + (−0.314 − 0.949i)4-s + (0.990 − 0.135i)5-s + (0.0339 − 0.999i)7-s + (−0.953 − 0.301i)8-s + (0.470 − 0.882i)10-s + (0.798 + 0.601i)11-s + (0.966 − 0.255i)13-s + (−0.790 − 0.612i)14-s + (−0.802 + 0.596i)16-s + (−0.909 + 0.415i)17-s + (0.202 − 0.979i)19-s + (−0.440 − 0.897i)20-s + (0.955 − 0.294i)22-s + (0.963 − 0.268i)25-s + (0.359 − 0.933i)26-s + ⋯
L(s)  = 1  + (0.585 − 0.810i)2-s + (−0.314 − 0.949i)4-s + (0.990 − 0.135i)5-s + (0.0339 − 0.999i)7-s + (−0.953 − 0.301i)8-s + (0.470 − 0.882i)10-s + (0.798 + 0.601i)11-s + (0.966 − 0.255i)13-s + (−0.790 − 0.612i)14-s + (−0.802 + 0.596i)16-s + (−0.909 + 0.415i)17-s + (0.202 − 0.979i)19-s + (−0.440 − 0.897i)20-s + (0.955 − 0.294i)22-s + (0.963 − 0.268i)25-s + (0.359 − 0.933i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $-0.624 - 0.780i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (1210, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ -0.624 - 0.780i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.507648077 - 3.137633455i\)
\(L(\frac12)\) \(\approx\) \(1.507648077 - 3.137633455i\)
\(L(1)\) \(\approx\) \(1.410597755 - 1.141049654i\)
\(L(1)\) \(\approx\) \(1.410597755 - 1.141049654i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.585 - 0.810i)T \)
5 \( 1 + (0.990 - 0.135i)T \)
7 \( 1 + (0.0339 - 0.999i)T \)
11 \( 1 + (0.798 + 0.601i)T \)
13 \( 1 + (0.966 - 0.255i)T \)
17 \( 1 + (-0.909 + 0.415i)T \)
19 \( 1 + (0.202 - 0.979i)T \)
31 \( 1 + (0.999 + 0.00679i)T \)
37 \( 1 + (0.670 - 0.742i)T \)
41 \( 1 + (-0.690 + 0.723i)T \)
43 \( 1 + (0.999 - 0.00679i)T \)
47 \( 1 + (0.997 + 0.0747i)T \)
53 \( 1 + (0.999 - 0.0407i)T \)
59 \( 1 + (0.981 + 0.189i)T \)
61 \( 1 + (-0.822 - 0.568i)T \)
67 \( 1 + (0.601 + 0.798i)T \)
71 \( 1 + (0.523 - 0.852i)T \)
73 \( 1 + (0.242 + 0.970i)T \)
79 \( 1 + (-0.993 - 0.115i)T \)
83 \( 1 + (0.833 + 0.552i)T \)
89 \( 1 + (-0.728 + 0.685i)T \)
97 \( 1 + (-0.0135 + 0.999i)T \)
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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

−17.81944661579521481281539567820, −17.21100001447662889533753885208, −16.587945184158197851796058121974, −15.90976652372223011730568468920, −15.364623935848256526440428408430, −14.59475419758579492634366927333, −13.99983095123996675159976579671, −13.56353825038053597956082424914, −12.92482649806706108167731616151, −12.03099344783389463785583722967, −11.62477914952160343617084552001, −10.75183586796437041692522831426, −9.76233998868425266244922936419, −8.948179189611040760752472802664, −8.76829791896124451186057450731, −7.93036251483840268626045467862, −6.838690860130586187948308937976, −6.33359904113014268525384415359, −5.858488914145612249413283569378, −5.28471439521913991722756174434, −4.36647288312385457003252237061, −3.61776466935285394230268908560, −2.78230010203632519759910841347, −2.11413738526108658609031482252, −1.078785942298413810292885571642, 0.80919247903785575945299807211, 1.2993743109929628732189564358, 2.17750769473687306228980512992, 2.85080373084573005421621501670, 3.92964856050660198536925232026, 4.28433501699070296122609483980, 5.07536514284097625183841754855, 5.91754738352646886682538133310, 6.56236772720074336455124106624, 7.05603570798424118833674929969, 8.350894164355582694754638559654, 9.06280747630926738150876769660, 9.64480103851512850422080754983, 10.28003690327997581240483740975, 10.946667819134510560574123081764, 11.36381358960218188731459261932, 12.376275722385124010685311599634, 12.98103864680411809578283711802, 13.62138833678994627090987671350, 13.83278292023354537024035852723, 14.69027641089556352431533005921, 15.319894157556377133154873753493, 16.12184639677340641521280473900, 17.04912445440162084236219642336, 17.64338129817008385160464409349

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