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Properties

Label 1-2011-2011.108-r1-0-0
Degree $1$
Conductor $2011$
Sign $0.298 - 0.954i$
Analytic cond. $216.111$
Root an. cond. $216.111$
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  + (0.837 + 0.546i)2-s + (0.0296 + 0.999i)3-s + (0.402 + 0.915i)4-s + (0.534 + 0.844i)5-s + (−0.521 + 0.853i)6-s + (0.252 + 0.967i)7-s + (−0.163 + 0.986i)8-s + (−0.998 + 0.0593i)9-s + (−0.0140 + 0.999i)10-s + (−0.581 + 0.813i)11-s + (−0.903 + 0.429i)12-s + (−0.537 + 0.843i)13-s + (−0.317 + 0.948i)14-s + (−0.828 + 0.559i)15-s + (−0.676 + 0.736i)16-s + (0.770 + 0.637i)17-s + ⋯
L(s)  = 1  + (0.837 + 0.546i)2-s + (0.0296 + 0.999i)3-s + (0.402 + 0.915i)4-s + (0.534 + 0.844i)5-s + (−0.521 + 0.853i)6-s + (0.252 + 0.967i)7-s + (−0.163 + 0.986i)8-s + (−0.998 + 0.0593i)9-s + (−0.0140 + 0.999i)10-s + (−0.581 + 0.813i)11-s + (−0.903 + 0.429i)12-s + (−0.537 + 0.843i)13-s + (−0.317 + 0.948i)14-s + (−0.828 + 0.559i)15-s + (−0.676 + 0.736i)16-s + (0.770 + 0.637i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2011\)
Sign: $0.298 - 0.954i$
Analytic conductor: \(216.111\)
Root analytic conductor: \(216.111\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2011} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2011,\ (1:\ ),\ 0.298 - 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-3.047463410 + 2.240538669i\)
\(L(\frac12)\) \(\approx\) \(-3.047463410 + 2.240538669i\)
\(L(1)\) \(\approx\) \(0.5457091288 + 1.701265114i\)
\(L(1)\) \(\approx\) \(0.5457091288 + 1.701265114i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2011 \( 1 \)
good2 \( 1 + (0.837 + 0.546i)T \)
3 \( 1 + (0.0296 + 0.999i)T \)
5 \( 1 + (0.534 + 0.844i)T \)
7 \( 1 + (0.252 + 0.967i)T \)
11 \( 1 + (-0.581 + 0.813i)T \)
13 \( 1 + (-0.537 + 0.843i)T \)
17 \( 1 + (0.770 + 0.637i)T \)
19 \( 1 + (-0.0203 + 0.999i)T \)
23 \( 1 + (-0.215 - 0.976i)T \)
29 \( 1 + (0.258 - 0.966i)T \)
31 \( 1 + (-0.405 - 0.914i)T \)
37 \( 1 + (0.996 + 0.0780i)T \)
41 \( 1 + (0.464 + 0.885i)T \)
43 \( 1 + (-0.227 + 0.973i)T \)
47 \( 1 + (0.976 - 0.214i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.965 + 0.259i)T \)
61 \( 1 + (-0.999 + 0.0125i)T \)
67 \( 1 + (0.999 + 0.0156i)T \)
71 \( 1 + (0.436 - 0.899i)T \)
73 \( 1 + (0.306 + 0.952i)T \)
79 \( 1 + (0.609 - 0.793i)T \)
83 \( 1 + (0.113 + 0.993i)T \)
89 \( 1 + (0.967 - 0.253i)T \)
97 \( 1 + (0.990 - 0.140i)T \)
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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

−19.48195405295097995406865700254, −18.49526374058022415152639275856, −17.82225013065114205973593441706, −17.067986148734514641793685176250, −16.30190172854465697268290601951, −15.43227012433549046678025969669, −14.27174134734164258093665640943, −13.821631657358114729295025015738, −13.3499408715621886397134508176, −12.61556156735980324638036784110, −12.093908331246168095130783347804, −11.085258831386691419617722902259, −10.53676845338255729183808387822, −9.55877696282605976214222770843, −8.68472937022274703680183026154, −7.6105194311029237251060220199, −7.12901836072488811483227739303, −5.956209517189475310900477118666, −5.36688008118126361219483592426, −4.793172774168790458683015465045, −3.49123499421955847519666748600, −2.79040519526758886994400438070, −1.82976789178579549397539740080, −0.83616445754028246173554193254, −0.54058921372455130585927562522, 2.15715263594385728049827479329, 2.442996933747159117458589215846, 3.485444821058049134709736249100, 4.370424250971242119629696525156, 5.03750048608580814853770985854, 6.01551935792407731051961469643, 6.23107716821118622930962028868, 7.6121630612247489848157103285, 8.13257335303420982937073976369, 9.28218555947843325002457551564, 9.90845921908608974796334958041, 10.72111394949581820773744571000, 11.61793332574208923197859179814, 12.19732537385490319788019848312, 13.08470721946357399540120966765, 14.161097822503727780438766350601, 14.68851550106089170190329672995, 14.97746112009745569863214581965, 15.739000378391423112722766867875, 16.67462974871391974921021534331, 17.08465339077976266738610715644, 18.12608579637744956198823230547, 18.66882883116715035589644754546, 19.8286354001144912322627901812, 20.79477956610386468259633246933

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