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Properties

Label 1-6003-6003.122-r0-0-0
Degree $1$
Conductor $6003$
Sign $0.144 - 0.989i$
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  + (0.155 − 0.987i)2-s + (−0.951 − 0.307i)4-s + (−0.0339 − 0.999i)5-s + (−0.390 + 0.920i)7-s + (−0.452 + 0.891i)8-s + (−0.992 − 0.122i)10-s + (0.973 − 0.229i)11-s + (−0.751 − 0.659i)13-s + (0.848 + 0.529i)14-s + (0.810 + 0.585i)16-s + (0.959 + 0.281i)17-s + (0.742 − 0.670i)19-s + (−0.275 + 0.961i)20-s + (−0.0747 − 0.997i)22-s + (−0.997 + 0.0679i)25-s + (−0.768 + 0.639i)26-s + ⋯
L(s)  = 1  + (0.155 − 0.987i)2-s + (−0.951 − 0.307i)4-s + (−0.0339 − 0.999i)5-s + (−0.390 + 0.920i)7-s + (−0.452 + 0.891i)8-s + (−0.992 − 0.122i)10-s + (0.973 − 0.229i)11-s + (−0.751 − 0.659i)13-s + (0.848 + 0.529i)14-s + (0.810 + 0.585i)16-s + (0.959 + 0.281i)17-s + (0.742 − 0.670i)19-s + (−0.275 + 0.961i)20-s + (−0.0747 − 0.997i)22-s + (−0.997 + 0.0679i)25-s + (−0.768 + 0.639i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6003\)    =    \(3^{2} \cdot 23 \cdot 29\)
Sign: $0.144 - 0.989i$
Analytic conductor: \(27.8778\)
Root analytic conductor: \(27.8778\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6003} (122, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6003,\ (0:\ ),\ 0.144 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.301412723 - 1.125624300i\)
\(L(\frac12)\) \(\approx\) \(1.301412723 - 1.125624300i\)
\(L(1)\) \(\approx\) \(0.8985807311 - 0.5907996346i\)
\(L(1)\) \(\approx\) \(0.8985807311 - 0.5907996346i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.155 - 0.987i)T \)
5 \( 1 + (-0.0339 - 0.999i)T \)
7 \( 1 + (-0.390 + 0.920i)T \)
11 \( 1 + (0.973 - 0.229i)T \)
13 \( 1 + (-0.751 - 0.659i)T \)
17 \( 1 + (0.959 + 0.281i)T \)
19 \( 1 + (0.742 - 0.670i)T \)
31 \( 1 + (0.923 + 0.384i)T \)
37 \( 1 + (0.182 - 0.983i)T \)
41 \( 1 + (0.981 + 0.189i)T \)
43 \( 1 + (-0.923 + 0.384i)T \)
47 \( 1 + (0.365 + 0.930i)T \)
53 \( 1 + (-0.714 - 0.699i)T \)
59 \( 1 + (-0.0475 + 0.998i)T \)
61 \( 1 + (0.855 + 0.517i)T \)
67 \( 1 + (0.973 + 0.229i)T \)
71 \( 1 + (0.862 + 0.505i)T \)
73 \( 1 + (0.0611 + 0.998i)T \)
79 \( 1 + (-0.912 - 0.409i)T \)
83 \( 1 + (-0.802 + 0.596i)T \)
89 \( 1 + (0.979 + 0.202i)T \)
97 \( 1 + (0.704 + 0.709i)T \)
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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

−17.57578452518079131811446562755, −17.11494023800377508623905500838, −16.63010930210234626243026651692, −15.93573586143085312574732059117, −15.1560399062428530063338170735, −14.517647688627154818214537299466, −14.01181907478060112049661917396, −13.75691314192437933869000976747, −12.66797740710820667773774396139, −11.98081976474867431759736612245, −11.40211130738354753342390847131, −10.25074080602160702230656376184, −9.79598482024509571400841966378, −9.39708728042070323968708218804, −8.200607923900099378963215335567, −7.596874292985760823439908304320, −7.014958098473833998807730789, −6.56590477313394715511158457728, −5.90041736760133140128983851188, −4.9402766440711817363479421862, −4.17868562781636182235788979156, −3.55899443641647627388890064008, −2.96936372680962098113458327001, −1.69165260028604592142879770194, −0.621838369268105412206511342036, 0.75830952739058419185594482143, 1.28318279416462014888387837704, 2.3415984774590224903089590181, 2.970344642221589400450727932962, 3.753792279779898879912499129230, 4.517827187046640954464049565525, 5.35800302460202451684237081743, 5.62741957347833448416593184205, 6.60978592502155790129255116531, 7.82105247214650457823528279284, 8.37784148142715416760118424567, 9.18116167599996015459886180132, 9.524414146042085530673683603864, 10.12082806538884717689102974533, 11.15365586814299326206432260120, 11.898913980599600546914464491993, 12.15308843766295047385099670924, 12.85784625333721387842445331586, 13.307328971168828672214658258337, 14.360829377539786317668411117491, 14.65009786454095583619355193230, 15.70158766901327321789772505787, 16.184175565796811992114306504971, 17.25301530035865325768246340642, 17.43964812607523446203107615140

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