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Properties

Label 1-6001-6001.1376-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.558 + 0.829i$
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  − i·2-s + (0.923 − 0.382i)3-s − 4-s + (−0.382 − 0.923i)5-s + (−0.382 − 0.923i)6-s + (0.382 − 0.923i)7-s + i·8-s + (0.707 − 0.707i)9-s + (−0.923 + 0.382i)10-s − i·11-s + (−0.923 + 0.382i)12-s + (−0.923 + 0.382i)13-s + (−0.923 − 0.382i)14-s + (−0.707 − 0.707i)15-s + 16-s + ⋯
L(s)  = 1  − i·2-s + (0.923 − 0.382i)3-s − 4-s + (−0.382 − 0.923i)5-s + (−0.382 − 0.923i)6-s + (0.382 − 0.923i)7-s + i·8-s + (0.707 − 0.707i)9-s + (−0.923 + 0.382i)10-s − i·11-s + (−0.923 + 0.382i)12-s + (−0.923 + 0.382i)13-s + (−0.923 − 0.382i)14-s + (−0.707 − 0.707i)15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.558 + 0.829i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (1376, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.558 + 0.829i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.7374712627 - 0.3927928093i\)
\(L(\frac12)\) \(\approx\) \(-0.7374712627 - 0.3927928093i\)
\(L(1)\) \(\approx\) \(0.5345187163 - 0.8990991172i\)
\(L(1)\) \(\approx\) \(0.5345187163 - 0.8990991172i\)

Euler product

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

−18.11561647913148980204292920848, −17.82548275184556535832747362226, −16.73441441277287188592018004917, −16.22111910554004167986818260970, −15.15568176211649845149032534583, −15.02427246865204379851383482609, −14.76830317675147771838302477250, −14.08710621205147466404706126012, −13.10265779883944434290321178892, −12.551842266438343530110794634544, −11.83964199490468820161227865433, −10.69479081168334724574133118142, −10.10997443436764970023646045622, −9.54073368584711578340480896183, −8.76780168830279415517804993462, −8.12804674327034028132489689894, −7.62665717696131596294339418231, −6.94868141906111688153894001248, −6.31583806688391178393579903076, −5.22122145973912706308491003242, −4.7476785478725695799369905890, −3.993419555295592340751669108672, −3.158076436747891578679588264827, −2.46020683341074243934187133806, −1.6855686084186456384864128573, 0.18726689024823762233959521289, 1.06227950976896059580215981178, 1.641325471768654274495522443189, 2.536476608866428100534069537028, 3.38344669371669766960356656295, 3.90171270996945562927899070885, 4.724801436840450520832470977, 5.12144792643387751134173464254, 6.47420546087910911804472154157, 7.31433400645957103677202025326, 8.07148800736784173912815508721, 8.49547724442116366959998904464, 9.14800031013126469472011227290, 9.776847885233832301339810009968, 10.520716205338632654125989441299, 11.38339772507926497102386871568, 11.85309583436965735346769587328, 12.66259973698590189942713237136, 13.20540400251772998871471682587, 13.784758142038950310297433888752, 14.1571116787934202997294233984, 15.08896109182008121241649010171, 15.71219039190600021072028486415, 16.90318523861092670694950195451, 17.12899974953944490164458841635

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