Properties

Label 1-373-373.59-r0-0-0
Degree $1$
Conductor $373$
Sign $0.903 + 0.428i$
Analytic cond. $1.73220$
Root an. cond. $1.73220$
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  + (0.905 − 0.425i)2-s + (−0.664 + 0.747i)3-s + (0.638 − 0.769i)4-s + (0.378 + 0.925i)5-s + (−0.283 + 0.959i)6-s + (0.528 + 0.848i)7-s + (0.250 − 0.968i)8-s + (−0.117 − 0.993i)9-s + (0.736 + 0.676i)10-s + (0.713 − 0.701i)11-s + (0.151 + 0.988i)12-s + (−0.250 − 0.968i)13-s + (0.839 + 0.543i)14-s + (−0.943 − 0.331i)15-s + (−0.184 − 0.982i)16-s + (0.347 + 0.937i)17-s + ⋯
L(s)  = 1  + (0.905 − 0.425i)2-s + (−0.664 + 0.747i)3-s + (0.638 − 0.769i)4-s + (0.378 + 0.925i)5-s + (−0.283 + 0.959i)6-s + (0.528 + 0.848i)7-s + (0.250 − 0.968i)8-s + (−0.117 − 0.993i)9-s + (0.736 + 0.676i)10-s + (0.713 − 0.701i)11-s + (0.151 + 0.988i)12-s + (−0.250 − 0.968i)13-s + (0.839 + 0.543i)14-s + (−0.943 − 0.331i)15-s + (−0.184 − 0.982i)16-s + (0.347 + 0.937i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.903 + 0.428i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.903 + 0.428i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.040048731 + 0.4598212518i\)
\(L(\frac12)\) \(\approx\) \(2.040048731 + 0.4598212518i\)
\(L(1)\) \(\approx\) \(1.642428759 + 0.1647126806i\)
\(L(1)\) \(\approx\) \(1.642428759 + 0.1647126806i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.905 - 0.425i)T \)
3 \( 1 + (-0.664 + 0.747i)T \)
5 \( 1 + (0.378 + 0.925i)T \)
7 \( 1 + (0.528 + 0.848i)T \)
11 \( 1 + (0.713 - 0.701i)T \)
13 \( 1 + (-0.250 - 0.968i)T \)
17 \( 1 + (0.347 + 0.937i)T \)
19 \( 1 + (0.758 + 0.651i)T \)
23 \( 1 + (-0.979 + 0.201i)T \)
29 \( 1 + (0.585 + 0.810i)T \)
31 \( 1 + (0.151 - 0.988i)T \)
37 \( 1 + (0.0168 + 0.999i)T \)
41 \( 1 + (0.688 + 0.724i)T \)
43 \( 1 + (-0.638 + 0.769i)T \)
47 \( 1 + (-0.890 - 0.455i)T \)
53 \( 1 + (0.117 - 0.993i)T \)
59 \( 1 + (0.409 - 0.912i)T \)
61 \( 1 + (0.664 - 0.747i)T \)
67 \( 1 + (0.612 - 0.790i)T \)
71 \( 1 + (-0.985 + 0.168i)T \)
73 \( 1 + (-0.801 - 0.598i)T \)
79 \( 1 + (-0.736 - 0.676i)T \)
83 \( 1 + (-0.117 + 0.993i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.347 - 0.937i)T \)
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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

−24.503824162705317863755458084612, −23.697855428305532221330224520065, −23.04225453938369567505169548746, −22.11860348071041005109561802146, −21.196280853462962862081626414392, −20.27507753997693967313544865125, −19.57944366632795540319164808872, −17.856082039600040520479361113031, −17.423739841517625406620170437, −16.54125777598640228538021840563, −15.95379809534490209466908678486, −14.16332059741908778526356689100, −13.96843168277709143583889082006, −12.9232530035757248676355909705, −11.92435155796506372233604326252, −11.60494080851815836626542835418, −10.10264591456995942485101771116, −8.72097423261295072332672568638, −7.46664967293140918398293054408, −6.95600962742756923898040248922, −5.794132845042081805191444694263, −4.80700625105096820790126768893, −4.22243707802780934813477827862, −2.27236120547017196083000688297, −1.22592955605023194835600348446, 1.52618475715407897772553816191, 2.98476341704162093817740143328, 3.71045722290444497107065196671, 5.07321782285705267997351261831, 5.88280705965606047344876097548, 6.41518818592594074400275422494, 8.083038141209282071735115851977, 9.69154569041440934715453057915, 10.266069545766635126846702090271, 11.32933641906099609885678543743, 11.77489245474411334583054080039, 12.864005956925944195943949183942, 14.25788827859719140891084973872, 14.736266923571152055056037933206, 15.5079337559286603473001703026, 16.50900538091877736686239976926, 17.72309322342285121749274450545, 18.47743523708527547022396055234, 19.55629091480461382899931964006, 20.65645113659946247388075177172, 21.56258681319859021404745125285, 22.03859040214026880863569845637, 22.54553775924879331633872302006, 23.555620691744619134142234692828, 24.524047542604735389173956136046

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