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Properties

Label 1-6001-6001.4579-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.954 - 0.299i$
Analytic cond. $27.8685$
Root an. cond. $27.8685$
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  − 2-s + (0.195 − 0.980i)3-s + 4-s + (−0.831 + 0.555i)5-s + (−0.195 + 0.980i)6-s + (0.555 − 0.831i)7-s − 8-s + (−0.923 − 0.382i)9-s + (0.831 − 0.555i)10-s + (0.382 + 0.923i)11-s + (0.195 − 0.980i)12-s + (0.831 + 0.555i)13-s + (−0.555 + 0.831i)14-s + (0.382 + 0.923i)15-s + 16-s + ⋯
L(s)  = 1  − 2-s + (0.195 − 0.980i)3-s + 4-s + (−0.831 + 0.555i)5-s + (−0.195 + 0.980i)6-s + (0.555 − 0.831i)7-s − 8-s + (−0.923 − 0.382i)9-s + (0.831 − 0.555i)10-s + (0.382 + 0.923i)11-s + (0.195 − 0.980i)12-s + (0.831 + 0.555i)13-s + (−0.555 + 0.831i)14-s + (0.382 + 0.923i)15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.954 - 0.299i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (4579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.954 - 0.299i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.208532272 - 0.1850394229i\)
\(L(\frac12)\) \(\approx\) \(1.208532272 - 0.1850394229i\)
\(L(1)\) \(\approx\) \(0.7591197917 - 0.1485650756i\)
\(L(1)\) \(\approx\) \(0.7591197917 - 0.1485650756i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (0.195 - 0.980i)T \)
5 \( 1 + (-0.831 + 0.555i)T \)
7 \( 1 + (0.555 - 0.831i)T \)
11 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 + (0.831 + 0.555i)T \)
19 \( 1 + (0.382 + 0.923i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (0.382 - 0.923i)T \)
31 \( 1 + (0.831 - 0.555i)T \)
37 \( 1 + (0.195 + 0.980i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (0.923 - 0.382i)T \)
47 \( 1 + (-0.382 - 0.923i)T \)
53 \( 1 + (-0.555 - 0.831i)T \)
59 \( 1 + (0.555 + 0.831i)T \)
61 \( 1 + (0.923 - 0.382i)T \)
67 \( 1 + (-0.555 + 0.831i)T \)
71 \( 1 + (-0.831 - 0.555i)T \)
73 \( 1 + (-0.923 - 0.382i)T \)
79 \( 1 + (-0.831 - 0.555i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.555 + 0.831i)T \)
97 \( 1 + (-0.382 - 0.923i)T \)
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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

−17.652876767120897590015056450069, −17.10395802528588514701886214906, −16.18652660830367490082321917579, −15.88768939496977217278520959469, −15.54917191702914799713713927779, −14.65355755081217277222906800605, −14.18809404385375402193713945541, −12.94468419621007912886441890740, −12.252377783636610612729033845248, −11.4252843357841648476529073425, −11.07452482197512374859417585047, −10.63303091491316067735114102118, −9.50881486753052915493794547133, −8.81867502227209395695760337791, −8.696589968597220235536042451470, −8.06647909012804534585920511620, −7.2466576146731803534676764856, −6.1737718522319679257671973040, −5.54890824526367681764462333455, −4.81917136414103306799765676763, −3.97194523671898185101473169512, −3.02178395183540116325732829745, −2.73022148680450971051300750715, −1.33133338483222336144490093450, −0.63328977557974069645021696526, 0.77247734232721205472329501326, 1.38776092584900859947860694192, 2.10188744208448427429262857221, 3.039813182575515290540256796934, 3.75974897335178814341271223693, 4.55950205988498179271389484915, 5.92262623108846984497322499396, 6.50763078831273794958143114393, 7.24606489473354353766404744812, 7.52124433980188638674902044237, 8.21800997656545370571149304415, 8.74531903324243868784734524302, 9.7835172905352009759266156776, 10.30239724683539759844618269785, 11.40488406860655160461028940896, 11.47470429435514121972015423310, 12.10173352796452377725827135287, 13.06222139230393718015241376302, 13.79373550414512363416806041883, 14.62278471882711841809647885851, 14.94270666961176467559688892252, 15.8389787290019659601025439421, 16.53853346688730527609824109865, 17.30737227931727130837939672618, 17.71021201105463763845103819597

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