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Properties

Label 1-373-373.65-r1-0-0
Degree $1$
Conductor $373$
Sign $0.906 + 0.422i$
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  + (−0.331 − 0.943i)2-s + (−0.283 − 0.959i)3-s + (−0.780 + 0.625i)4-s + (−0.514 − 0.857i)5-s + (−0.810 + 0.585i)6-s + (0.612 − 0.790i)7-s + (0.848 + 0.528i)8-s + (−0.839 + 0.543i)9-s + (−0.638 + 0.769i)10-s + (−0.882 − 0.470i)11-s + (0.820 + 0.571i)12-s + (−0.528 − 0.848i)13-s + (−0.948 − 0.315i)14-s + (−0.676 + 0.736i)15-s + (0.217 − 0.975i)16-s + (−0.151 − 0.988i)17-s + ⋯
L(s)  = 1  + (−0.331 − 0.943i)2-s + (−0.283 − 0.959i)3-s + (−0.780 + 0.625i)4-s + (−0.514 − 0.857i)5-s + (−0.810 + 0.585i)6-s + (0.612 − 0.790i)7-s + (0.848 + 0.528i)8-s + (−0.839 + 0.543i)9-s + (−0.638 + 0.769i)10-s + (−0.882 − 0.470i)11-s + (0.820 + 0.571i)12-s + (−0.528 − 0.848i)13-s + (−0.948 − 0.315i)14-s + (−0.676 + 0.736i)15-s + (0.217 − 0.975i)16-s + (−0.151 − 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.906 + 0.422i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.906 + 0.422i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4698982924 - 0.1040600681i\)
\(L(\frac12)\) \(\approx\) \(-0.4698982924 - 0.1040600681i\)
\(L(1)\) \(\approx\) \(0.2204958246 - 0.5492632779i\)
\(L(1)\) \(\approx\) \(0.2204958246 - 0.5492632779i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.331 - 0.943i)T \)
3 \( 1 + (-0.283 - 0.959i)T \)
5 \( 1 + (-0.514 - 0.857i)T \)
7 \( 1 + (0.612 - 0.790i)T \)
11 \( 1 + (-0.882 - 0.470i)T \)
13 \( 1 + (-0.528 - 0.848i)T \)
17 \( 1 + (-0.151 - 0.988i)T \)
19 \( 1 + (0.299 - 0.954i)T \)
23 \( 1 + (-0.724 + 0.688i)T \)
29 \( 1 + (-0.117 - 0.993i)T \)
31 \( 1 + (-0.820 + 0.571i)T \)
37 \( 1 + (0.557 - 0.830i)T \)
41 \( 1 + (-0.994 + 0.101i)T \)
43 \( 1 + (0.625 + 0.780i)T \)
47 \( 1 + (-0.747 - 0.664i)T \)
53 \( 1 + (-0.543 + 0.839i)T \)
59 \( 1 + (0.801 - 0.598i)T \)
61 \( 1 + (0.959 - 0.283i)T \)
67 \( 1 + (-0.485 + 0.874i)T \)
71 \( 1 + (0.931 - 0.363i)T \)
73 \( 1 + (0.890 - 0.455i)T \)
79 \( 1 + (-0.769 - 0.638i)T \)
83 \( 1 + (-0.839 - 0.543i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.988 - 0.151i)T \)
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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

−25.64503653563011749627210346740, −24.04151093159660996385552173952, −23.711657567983631137096593561631, −22.453382758494093454414773604563, −22.089345358051726536972699961433, −21.028239041376435920241838304040, −19.84133022378299430827021531728, −18.63775024327608137824414185186, −18.19311491352034792541476346154, −17.12223988860715199538571732396, −16.19955650169109444123896444578, −15.43994368623032930140856335209, −14.70423670702919280700165013939, −14.30730431477411189767942859890, −12.540652348978703826420747144638, −11.439749805923889709733274794710, −10.47083435938694903946375829696, −9.77215635387086384410163918126, −8.57519004376729434693344315099, −7.82973488901353155100270734139, −6.61498539178841926208311187436, −5.638264525187259689010489752467, −4.71769144375882482572273567393, −3.7421678969148163897295994888, −2.12104227525852214909739291579, 0.2279163667791936324339593462, 0.76308392298114414530222270209, 2.060666034865505954872860873739, 3.27205258688541676604128122493, 4.71718653479944190496076733764, 5.399119864983498563362024986994, 7.44470168598888921403984086021, 7.785323219287596035093000712107, 8.75285678691613143435386295199, 10.01798856203960447747985436572, 11.22878839995012943472794176651, 11.61218348193115763253789404497, 12.80192345771325728574602391659, 13.28647831881474615638307449923, 14.15381536040641661807038324199, 15.841348457171508401251222368144, 16.83244132754627060133550972957, 17.67702490418235346046675778087, 18.20100441598312608291830217261, 19.40072501561093972790342256579, 20.021521488886317766946411548895, 20.596965419397307919377230556399, 21.71000437264902068101684644625, 22.851543413453152285625479354929, 23.56138919721212723634296358517

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