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Properties

Label 1-101-101.98-r1-0-0
Degree $1$
Conductor $101$
Sign $-0.358 - 0.933i$
Analytic cond. $10.8539$
Root an. cond. $10.8539$
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  + (0.368 + 0.929i)2-s + (0.770 + 0.637i)3-s + (−0.728 + 0.684i)4-s + (−0.929 − 0.368i)5-s + (−0.309 + 0.951i)6-s + (−0.248 − 0.968i)7-s + (−0.904 − 0.425i)8-s + (0.187 + 0.982i)9-s − i·10-s + (−0.982 + 0.187i)11-s + (−0.998 + 0.0627i)12-s + (−0.968 − 0.248i)13-s + (0.809 − 0.587i)14-s + (−0.481 − 0.876i)15-s + (0.0627 − 0.998i)16-s + (−0.309 − 0.951i)17-s + ⋯
L(s)  = 1  + (0.368 + 0.929i)2-s + (0.770 + 0.637i)3-s + (−0.728 + 0.684i)4-s + (−0.929 − 0.368i)5-s + (−0.309 + 0.951i)6-s + (−0.248 − 0.968i)7-s + (−0.904 − 0.425i)8-s + (0.187 + 0.982i)9-s − i·10-s + (−0.982 + 0.187i)11-s + (−0.998 + 0.0627i)12-s + (−0.968 − 0.248i)13-s + (0.809 − 0.587i)14-s + (−0.481 − 0.876i)15-s + (0.0627 − 0.998i)16-s + (−0.309 − 0.951i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.358 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(101\)
Sign: $-0.358 - 0.933i$
Analytic conductor: \(10.8539\)
Root analytic conductor: \(10.8539\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{101} (98, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 101,\ (1:\ ),\ -0.358 - 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1713212057 + 0.2493077621i\)
\(L(\frac12)\) \(\approx\) \(-0.1713212057 + 0.2493077621i\)
\(L(1)\) \(\approx\) \(0.6884265386 + 0.5314435987i\)
\(L(1)\) \(\approx\) \(0.6884265386 + 0.5314435987i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad101 \( 1 \)
good2 \( 1 + (0.368 + 0.929i)T \)
3 \( 1 + (0.770 + 0.637i)T \)
5 \( 1 + (-0.929 - 0.368i)T \)
7 \( 1 + (-0.248 - 0.968i)T \)
11 \( 1 + (-0.982 + 0.187i)T \)
13 \( 1 + (-0.968 - 0.248i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.0627 + 0.998i)T \)
23 \( 1 + (-0.535 + 0.844i)T \)
29 \( 1 + (-0.248 + 0.968i)T \)
31 \( 1 + (0.968 - 0.248i)T \)
37 \( 1 + (-0.637 - 0.770i)T \)
41 \( 1 + (-0.951 - 0.309i)T \)
43 \( 1 + (0.992 - 0.125i)T \)
47 \( 1 + (0.992 + 0.125i)T \)
53 \( 1 + (-0.684 + 0.728i)T \)
59 \( 1 + (-0.998 - 0.0627i)T \)
61 \( 1 + (-0.684 - 0.728i)T \)
67 \( 1 + (-0.770 + 0.637i)T \)
71 \( 1 + (-0.637 + 0.770i)T \)
73 \( 1 + (-0.844 - 0.535i)T \)
79 \( 1 + (0.535 + 0.844i)T \)
83 \( 1 + (0.844 - 0.535i)T \)
89 \( 1 + (0.998 - 0.0627i)T \)
97 \( 1 + (0.728 - 0.684i)T \)
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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

−28.986735375909148865987233285116, −28.219155669418339780379762177888, −26.79434676427674305802013275240, −26.10236073112547681028846614736, −24.379287702302668977299629617707, −23.836486253289024710047760593437, −22.53553197577379305915206475798, −21.54519367486899903292157830724, −20.358866664597117226231449780969, −19.27313735284500671351614627821, −18.927533392838164014899808932909, −17.8017242582582227429659225465, −15.48380118776246805465260852072, −14.91649081032910562142325618053, −13.58764906039322456269221545834, −12.4936888823575648345253675426, −11.81025341014693853972136207620, −10.35372317522832672954193715179, −8.955321428551191582760643894700, −7.95843332588837469809850637245, −6.34581227174576367139648179670, −4.5801618190040209236003515486, −3.06184741513054872555942836148, −2.28142249781666237640252886146, −0.10128410366150054797997830425, 3.126420716838632109860606604575, 4.23082970292227586617051500825, 5.17392847172358930076929142820, 7.39752530309884137694396487631, 7.79655306993418932759126831518, 9.20869278111105258718356749854, 10.42078702687472173749962313860, 12.258240375911065938997386186055, 13.440337210661004006484142583731, 14.39109023617925610210148699982, 15.58240324715592285464987332760, 16.11836496175399143801406687186, 17.22096062079654763629908636256, 18.8323979963552546944625335505, 20.08504135723608284045676261608, 20.84680707310130200955173161381, 22.266645351182361382457328633084, 23.18520244002722742351888822871, 24.15473956651765188919604124953, 25.204177025887435455630062254558, 26.35667798473457890573832510910, 26.92434054780919676930177130673, 27.74154533149091822205088413563, 29.538261177088505289699180697442, 30.88791436969998892352216486178

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