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Properties

Label 1-373-373.114-r0-0-0
Degree $1$
Conductor $373$
Sign $0.928 + 0.372i$
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  + (−0.217 − 0.975i)2-s + (0.409 + 0.912i)3-s + (−0.905 + 0.425i)4-s + (0.557 − 0.830i)5-s + (0.801 − 0.598i)6-s + (−0.874 − 0.485i)7-s + (0.612 + 0.790i)8-s + (−0.664 + 0.747i)9-s + (−0.931 − 0.363i)10-s + (0.378 + 0.925i)11-s + (−0.758 − 0.651i)12-s + (−0.612 + 0.790i)13-s + (−0.283 + 0.959i)14-s + (0.985 + 0.168i)15-s + (0.638 − 0.769i)16-s + (0.820 + 0.571i)17-s + ⋯
L(s)  = 1  + (−0.217 − 0.975i)2-s + (0.409 + 0.912i)3-s + (−0.905 + 0.425i)4-s + (0.557 − 0.830i)5-s + (0.801 − 0.598i)6-s + (−0.874 − 0.485i)7-s + (0.612 + 0.790i)8-s + (−0.664 + 0.747i)9-s + (−0.931 − 0.363i)10-s + (0.378 + 0.925i)11-s + (−0.758 − 0.651i)12-s + (−0.612 + 0.790i)13-s + (−0.283 + 0.959i)14-s + (0.985 + 0.168i)15-s + (0.638 − 0.769i)16-s + (0.820 + 0.571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.928 + 0.372i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.928 + 0.372i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.077751894 + 0.2080272175i\)
\(L(\frac12)\) \(\approx\) \(1.077751894 + 0.2080272175i\)
\(L(1)\) \(\approx\) \(0.9745663038 - 0.07914337938i\)
\(L(1)\) \(\approx\) \(0.9745663038 - 0.07914337938i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.217 - 0.975i)T \)
3 \( 1 + (0.409 + 0.912i)T \)
5 \( 1 + (0.557 - 0.830i)T \)
7 \( 1 + (-0.874 - 0.485i)T \)
11 \( 1 + (0.378 + 0.925i)T \)
13 \( 1 + (-0.612 + 0.790i)T \)
17 \( 1 + (0.820 + 0.571i)T \)
19 \( 1 + (-0.347 + 0.937i)T \)
23 \( 1 + (0.994 - 0.101i)T \)
29 \( 1 + (0.890 + 0.455i)T \)
31 \( 1 + (-0.758 + 0.651i)T \)
37 \( 1 + (-0.713 - 0.701i)T \)
41 \( 1 + (0.918 + 0.394i)T \)
43 \( 1 + (0.905 - 0.425i)T \)
47 \( 1 + (0.972 + 0.234i)T \)
53 \( 1 + (0.664 + 0.747i)T \)
59 \( 1 + (-0.839 + 0.543i)T \)
61 \( 1 + (-0.409 - 0.912i)T \)
67 \( 1 + (0.440 + 0.897i)T \)
71 \( 1 + (0.0843 + 0.996i)T \)
73 \( 1 + (-0.315 + 0.948i)T \)
79 \( 1 + (0.931 + 0.363i)T \)
83 \( 1 + (-0.664 - 0.747i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.820 - 0.571i)T \)
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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

−24.83454969807946049813399009178, −23.9143885373677226141920749689, −22.81221438154390537674752535524, −22.36638293076390360472330746945, −21.26536455428771246143449418616, −19.58511845451138747392139148632, −19.11715651915953369283953844025, −18.400798371726683653456245635251, −17.545902096986144520790268522605, −16.75782331013892613469109161639, −15.47341571109084744828965698502, −14.7744007619448814234770332167, −13.84783102638958556985349181217, −13.25511744718678418553973696141, −12.236234927320399035937408900610, −10.7861503029441548009643511672, −9.56020204272663824754555021109, −8.9504691968207863512663567895, −7.739086040071483727223819992302, −6.91591476714106084177926096949, −6.165768447954637329011233920814, −5.39690531458997973591337228876, −3.38599949325084524939382001655, −2.57328750186299051353340879778, −0.73726555753746669690726339307, 1.42446155298704311812131633335, 2.60973761720396180482934150316, 3.85970187372273917999913754177, 4.50556526807585058037164566092, 5.62126607199117535383983377878, 7.32784932077754186312226184822, 8.690033633587441859402988715964, 9.35592401125995665119224023565, 10.01413379254036616005857086782, 10.70856208239449690843086664709, 12.28699125010682429103691650945, 12.678078321347425969714503394750, 13.96764758832943893011430096225, 14.51724985031885392718461428562, 16.06073442892953974299857460853, 16.91476219083170977689286819029, 17.30170226192101092250983060995, 18.885207002217352724158981337623, 19.6935317613434876192889303954, 20.26869498105275948235909822667, 21.181214200562065469965522108064, 21.6483564699576980868626950013, 22.72209634526689060369565577799, 23.406037458548406915219999220660, 25.06410526657059253479805863978

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