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Properties

Label 1-373-373.48-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.978 - 0.204i$
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  + (−0.585 − 0.810i)2-s + (0.470 − 0.882i)3-s + (−0.315 + 0.948i)4-s + (0.931 − 0.363i)5-s + (−0.990 + 0.134i)6-s + (0.347 − 0.937i)7-s + (0.954 − 0.299i)8-s + (−0.557 − 0.830i)9-s + (−0.839 − 0.543i)10-s + (−0.736 + 0.676i)11-s + (0.688 + 0.724i)12-s + (−0.954 − 0.299i)13-s + (−0.963 + 0.266i)14-s + (0.117 − 0.993i)15-s + (−0.801 − 0.598i)16-s + (0.979 − 0.201i)17-s + ⋯
L(s)  = 1  + (−0.585 − 0.810i)2-s + (0.470 − 0.882i)3-s + (−0.315 + 0.948i)4-s + (0.931 − 0.363i)5-s + (−0.990 + 0.134i)6-s + (0.347 − 0.937i)7-s + (0.954 − 0.299i)8-s + (−0.557 − 0.830i)9-s + (−0.839 − 0.543i)10-s + (−0.736 + 0.676i)11-s + (0.688 + 0.724i)12-s + (−0.954 − 0.299i)13-s + (−0.963 + 0.266i)14-s + (0.117 − 0.993i)15-s + (−0.801 − 0.598i)16-s + (0.979 − 0.201i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.978 - 0.204i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.978 - 0.204i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1135847903 - 1.098656387i\)
\(L(\frac12)\) \(\approx\) \(0.1135847903 - 1.098656387i\)
\(L(1)\) \(\approx\) \(0.6343867051 - 0.7336504117i\)
\(L(1)\) \(\approx\) \(0.6343867051 - 0.7336504117i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.585 - 0.810i)T \)
3 \( 1 + (0.470 - 0.882i)T \)
5 \( 1 + (0.931 - 0.363i)T \)
7 \( 1 + (0.347 - 0.937i)T \)
11 \( 1 + (-0.736 + 0.676i)T \)
13 \( 1 + (-0.954 - 0.299i)T \)
17 \( 1 + (0.979 - 0.201i)T \)
19 \( 1 + (-0.918 - 0.394i)T \)
23 \( 1 + (-0.528 + 0.848i)T \)
29 \( 1 + (0.0168 - 0.999i)T \)
31 \( 1 + (0.688 - 0.724i)T \)
37 \( 1 + (0.0843 + 0.996i)T \)
41 \( 1 + (-0.612 - 0.790i)T \)
43 \( 1 + (0.315 - 0.948i)T \)
47 \( 1 + (0.713 - 0.701i)T \)
53 \( 1 + (0.557 - 0.830i)T \)
59 \( 1 + (0.857 + 0.514i)T \)
61 \( 1 + (-0.470 + 0.882i)T \)
67 \( 1 + (-0.151 + 0.988i)T \)
71 \( 1 + (-0.664 + 0.747i)T \)
73 \( 1 + (0.997 + 0.0675i)T \)
79 \( 1 + (0.839 + 0.543i)T \)
83 \( 1 + (-0.557 + 0.830i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.979 + 0.201i)T \)
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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

−25.148698068399065907782702362203, −24.56838929956242119846894564346, −23.39541861545513501071386279568, −22.21109726250550326221540540364, −21.55057176442238652521505110335, −20.865072084634713845955669125, −19.49424573853085467617313584849, −18.74629119161009557624200975449, −17.949740944234166312059626745980, −16.869086975573570647937008060941, −16.2542679085031212797689686508, −15.16911505242119076153446857353, −14.48770629723356234888094966353, −13.991069933541900906175020050561, −12.55846148176000239818513915616, −10.90183817748109665946122984120, −10.26469909707482477015659660784, −9.415755381820611160752027269849, −8.58941283930694355205405967476, −7.79014834901135995901481401852, −6.28689663888790962165970615673, −5.48502777938685772419909812989, −4.7032661216017876561429449336, −2.896434234508982940530764259254, −1.93320231576363924331923606991, 0.747380272601290617812777204752, 1.94242544207079756676943136680, 2.63215687563997893628021745822, 4.10473353341784341208883490915, 5.377861992013727723715667280597, 6.97997592586022350515540481235, 7.71346210733644249613392518880, 8.55811774609130859073198731156, 9.88386253932652612241037665295, 10.166224017983989533712409429965, 11.68296812412427027850725624130, 12.52331162615819404250936755696, 13.370881765468324859561971966138, 13.87120535053903569529326434429, 15.12457886135328986879045765635, 16.89703438236570605417757373813, 17.32491357825160847952014614327, 18.01461835142043653273037398467, 19.00392524903157157321807634801, 19.83835797055904067696278656651, 20.685598313458186974226451598216, 21.064086524579380098282110722252, 22.34547240556843982141008679999, 23.418840012290394922259154013828, 24.26233022652990042915066305886

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