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Properties

Label 1-373-373.32-r1-0-0
Degree $1$
Conductor $373$
Sign $-0.993 + 0.114i$
Analytic cond. $40.0844$
Root an. cond. $40.0844$
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  + (0.996 + 0.0843i)2-s + (0.315 + 0.948i)3-s + (0.985 + 0.168i)4-s + (−0.134 + 0.990i)5-s + (0.234 + 0.972i)6-s + (−0.528 − 0.848i)7-s + (0.968 + 0.250i)8-s + (−0.801 + 0.598i)9-s + (−0.217 + 0.975i)10-s + (−0.266 + 0.963i)11-s + (0.151 + 0.988i)12-s + (0.250 + 0.968i)13-s + (−0.455 − 0.890i)14-s + (−0.982 + 0.184i)15-s + (0.943 + 0.331i)16-s + (−0.347 − 0.937i)17-s + ⋯
L(s)  = 1  + (0.996 + 0.0843i)2-s + (0.315 + 0.948i)3-s + (0.985 + 0.168i)4-s + (−0.134 + 0.990i)5-s + (0.234 + 0.972i)6-s + (−0.528 − 0.848i)7-s + (0.968 + 0.250i)8-s + (−0.801 + 0.598i)9-s + (−0.217 + 0.975i)10-s + (−0.266 + 0.963i)11-s + (0.151 + 0.988i)12-s + (0.250 + 0.968i)13-s + (−0.455 − 0.890i)14-s + (−0.982 + 0.184i)15-s + (0.943 + 0.331i)16-s + (−0.347 − 0.937i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.993 + 0.114i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ -0.993 + 0.114i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1662141081 + 2.887899072i\)
\(L(\frac12)\) \(\approx\) \(0.1662141081 + 2.887899072i\)
\(L(1)\) \(\approx\) \(1.409758549 + 1.127068175i\)
\(L(1)\) \(\approx\) \(1.409758549 + 1.127068175i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.996 + 0.0843i)T \)
3 \( 1 + (0.315 + 0.948i)T \)
5 \( 1 + (-0.134 + 0.990i)T \)
7 \( 1 + (-0.528 - 0.848i)T \)
11 \( 1 + (-0.266 + 0.963i)T \)
13 \( 1 + (0.250 + 0.968i)T \)
17 \( 1 + (-0.347 - 0.937i)T \)
19 \( 1 + (-0.651 + 0.758i)T \)
23 \( 1 + (-0.201 - 0.979i)T \)
29 \( 1 + (0.409 - 0.912i)T \)
31 \( 1 + (-0.151 + 0.988i)T \)
37 \( 1 + (-0.857 + 0.514i)T \)
41 \( 1 + (0.688 + 0.724i)T \)
43 \( 1 + (-0.168 + 0.985i)T \)
47 \( 1 + (0.543 + 0.839i)T \)
53 \( 1 + (0.598 - 0.801i)T \)
59 \( 1 + (-0.585 - 0.810i)T \)
61 \( 1 + (0.948 - 0.315i)T \)
67 \( 1 + (-0.790 - 0.612i)T \)
71 \( 1 + (-0.638 - 0.769i)T \)
73 \( 1 + (-0.117 + 0.993i)T \)
79 \( 1 + (0.975 + 0.217i)T \)
83 \( 1 + (-0.801 - 0.598i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.937 + 0.347i)T \)
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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

−23.95648047949321361791132768164, −23.39989394712519341640647601126, −22.19933970778623113523466806278, −21.42767535429443544259148667546, −20.468128679627833690544075463686, −19.57832088288673806555540422058, −19.135909830903942387281668833464, −17.79160711350457245333369874994, −16.755847616983722516487249439944, −15.658315828905651551172889439434, −15.11739767544382016960453819585, −13.71070327439149035520034459880, −13.12538981103168372224068666347, −12.53152286885972870841935892411, −11.74105869311019423007773520656, −10.63044717503397912664060753162, −8.95592933439221457877804810186, −8.314548348938041810877584494580, −7.14879863006297790631251167009, −5.80937108712941265768478477812, −5.59160883474691615012677495151, −3.89474268687452756604780541924, −2.89656187702889086890417656403, −1.82338063974057441143930649764, −0.49060680134513562751919804670, 2.16065769861584231365768722983, 3.11250753816333438330606152807, 4.10637570240747169223369399054, 4.69689030179169617160306981754, 6.27129852163581760112175750745, 6.96586401613399868503937336513, 8.030977197710684609463283169680, 9.66556697471620130271888517388, 10.41725224652831967416110331457, 11.14112951117625991792676352783, 12.21022130678750382784601528237, 13.50618774174794732427340366292, 14.22936892887661047636349783100, 14.82783025082148086340350725994, 15.86165343634914154131091901933, 16.382193299506997698474573213799, 17.514217129191009296689088676841, 18.983827685819227981915272586659, 19.83718733569953990356011348569, 20.677576566800181213214823098962, 21.33731075778284980970543225782, 22.43244293570490808988343079422, 22.87619006410142661601387501277, 23.49142676956669035779500288434, 24.9396776722485544980790645226

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