Properties

Label 1-2001-2001.242-r0-0-0
Degree $1$
Conductor $2001$
Sign $-0.850 - 0.525i$
Analytic cond. $9.29260$
Root an. cond. $9.29260$
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  + (0.639 + 0.768i)2-s + (−0.182 + 0.983i)4-s + (−0.992 + 0.122i)5-s + (0.685 + 0.728i)7-s + (−0.872 + 0.488i)8-s + (−0.728 − 0.685i)10-s + (0.202 + 0.979i)11-s + (−0.996 − 0.0815i)13-s + (−0.122 + 0.992i)14-s + (−0.933 − 0.359i)16-s + (0.755 + 0.654i)17-s + (0.983 + 0.182i)19-s + (0.0611 − 0.998i)20-s + (−0.623 + 0.781i)22-s + (0.970 − 0.242i)25-s + (−0.574 − 0.818i)26-s + ⋯
L(s)  = 1  + (0.639 + 0.768i)2-s + (−0.182 + 0.983i)4-s + (−0.992 + 0.122i)5-s + (0.685 + 0.728i)7-s + (−0.872 + 0.488i)8-s + (−0.728 − 0.685i)10-s + (0.202 + 0.979i)11-s + (−0.996 − 0.0815i)13-s + (−0.122 + 0.992i)14-s + (−0.933 − 0.359i)16-s + (0.755 + 0.654i)17-s + (0.983 + 0.182i)19-s + (0.0611 − 0.998i)20-s + (−0.623 + 0.781i)22-s + (0.970 − 0.242i)25-s + (−0.574 − 0.818i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2001\)    =    \(3 \cdot 23 \cdot 29\)
Sign: $-0.850 - 0.525i$
Analytic conductor: \(9.29260\)
Root analytic conductor: \(9.29260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2001} (242, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2001,\ (0:\ ),\ -0.850 - 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3925954031 + 1.382899355i\)
\(L(\frac12)\) \(\approx\) \(-0.3925954031 + 1.382899355i\)
\(L(1)\) \(\approx\) \(0.7924477771 + 0.8766055614i\)
\(L(1)\) \(\approx\) \(0.7924477771 + 0.8766055614i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.639 + 0.768i)T \)
5 \( 1 + (-0.992 + 0.122i)T \)
7 \( 1 + (0.685 + 0.728i)T \)
11 \( 1 + (0.202 + 0.979i)T \)
13 \( 1 + (-0.996 - 0.0815i)T \)
17 \( 1 + (0.755 + 0.654i)T \)
19 \( 1 + (0.983 + 0.182i)T \)
31 \( 1 + (-0.162 + 0.986i)T \)
37 \( 1 + (-0.940 - 0.339i)T \)
41 \( 1 + (0.540 + 0.841i)T \)
43 \( 1 + (0.162 + 0.986i)T \)
47 \( 1 + (-0.974 - 0.222i)T \)
53 \( 1 + (-0.557 + 0.830i)T \)
59 \( 1 + (0.142 + 0.989i)T \)
61 \( 1 + (-0.925 - 0.377i)T \)
67 \( 1 + (0.979 + 0.202i)T \)
71 \( 1 + (0.794 + 0.607i)T \)
73 \( 1 + (0.396 - 0.917i)T \)
79 \( 1 + (-0.359 - 0.933i)T \)
83 \( 1 + (-0.101 - 0.994i)T \)
89 \( 1 + (0.670 - 0.742i)T \)
97 \( 1 + (0.320 - 0.947i)T \)
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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

−19.618030095594914620134980297860, −19.0255261357552787038807675569, −18.38516891442951544992550314460, −17.333374972507645417302123947717, −16.5329671069213424057253638501, −15.7641407690750509092134661492, −14.96343651025809841500232022299, −14.11889155274349696060855955147, −13.874119212815630265593314827521, −12.74666014446677842302562221034, −12.004050452837204255102930162259, −11.44565518740104175265947819854, −10.93574455492102053600334477869, −9.988071627293232318565014338953, −9.233676526729755343351788137093, −8.17985246989812933100545497950, −7.46690168406671995436794747768, −6.66218726806170350462720513872, −5.29625503372486868428222810004, −4.987634435043454055071558590, −3.889207902047775786336904458062, −3.44397349078484510570539855616, −2.43407344972219092465583229917, −1.19424136309925037093266219495, −0.428869114741153439060923646987, 1.56249812614332396559922798613, 2.76340918798912368955676006194, 3.4871661557474505941822170773, 4.56057395292864743880730951506, 4.95515695991167211717954133522, 5.86009117849831913237601234261, 6.912114085560713298770745805896, 7.60083360939354102875174084398, 8.034878865089559553651716447, 8.95378354880127283219665911860, 9.82621130646388271563905343316, 10.999648997398235002125892099487, 11.954621086857637786273838549043, 12.20247672472000847643062315471, 12.85429027422706346727469487601, 14.212555614815794027926076890740, 14.616642226257796971857620612376, 15.156849686851539650127208257638, 15.849444406062454235947924793776, 16.555538496439586065418146219572, 17.468132846081911967108869679370, 17.98665206322478707445189930542, 18.8323320100233380499941587707, 19.7624788367492011280833809397, 20.39935278939668333128896993073

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