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Properties

Label 1-373-373.117-r0-0-0
Degree $1$
Conductor $373$
Sign $0.250 + 0.968i$
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  + (−0.990 − 0.134i)2-s + (0.736 + 0.676i)3-s + (0.963 + 0.266i)4-s + (0.664 + 0.747i)5-s + (−0.638 − 0.769i)6-s + (−0.0506 + 0.998i)7-s + (−0.918 − 0.394i)8-s + (0.0843 + 0.996i)9-s + (−0.557 − 0.830i)10-s + (0.117 − 0.993i)11-s + (0.528 + 0.848i)12-s + (0.918 − 0.394i)13-s + (0.184 − 0.982i)14-s + (−0.0168 + 0.999i)15-s + (0.857 + 0.514i)16-s + (−0.250 − 0.968i)17-s + ⋯
L(s)  = 1  + (−0.990 − 0.134i)2-s + (0.736 + 0.676i)3-s + (0.963 + 0.266i)4-s + (0.664 + 0.747i)5-s + (−0.638 − 0.769i)6-s + (−0.0506 + 0.998i)7-s + (−0.918 − 0.394i)8-s + (0.0843 + 0.996i)9-s + (−0.557 − 0.830i)10-s + (0.117 − 0.993i)11-s + (0.528 + 0.848i)12-s + (0.918 − 0.394i)13-s + (0.184 − 0.982i)14-s + (−0.0168 + 0.999i)15-s + (0.857 + 0.514i)16-s + (−0.250 − 0.968i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.250 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.250 + 0.968i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (117, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ 0.250 + 0.968i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.019103610 + 0.7886859909i\)
\(L(\frac12)\) \(\approx\) \(1.019103610 + 0.7886859909i\)
\(L(1)\) \(\approx\) \(0.9731597049 + 0.3892179999i\)
\(L(1)\) \(\approx\) \(0.9731597049 + 0.3892179999i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.990 - 0.134i)T \)
3 \( 1 + (0.736 + 0.676i)T \)
5 \( 1 + (0.664 + 0.747i)T \)
7 \( 1 + (-0.0506 + 0.998i)T \)
11 \( 1 + (0.117 - 0.993i)T \)
13 \( 1 + (0.918 - 0.394i)T \)
17 \( 1 + (-0.250 - 0.968i)T \)
19 \( 1 + (0.874 + 0.485i)T \)
23 \( 1 + (0.954 + 0.299i)T \)
29 \( 1 + (0.780 + 0.625i)T \)
31 \( 1 + (0.528 - 0.848i)T \)
37 \( 1 + (-0.972 + 0.234i)T \)
41 \( 1 + (0.347 - 0.937i)T \)
43 \( 1 + (-0.963 - 0.266i)T \)
47 \( 1 + (-0.943 - 0.331i)T \)
53 \( 1 + (-0.0843 + 0.996i)T \)
59 \( 1 + (-0.931 + 0.363i)T \)
61 \( 1 + (-0.736 - 0.676i)T \)
67 \( 1 + (-0.979 + 0.201i)T \)
71 \( 1 + (-0.713 - 0.701i)T \)
73 \( 1 + (-0.905 + 0.425i)T \)
79 \( 1 + (0.557 + 0.830i)T \)
83 \( 1 + (0.0843 - 0.996i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.250 + 0.968i)T \)
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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

−24.71089940135779707876011776922, −23.8490236412501877978614373016, −23.103379202701561836816781677753, −21.19526256725273103114253292424, −20.718647292343074050522759296956, −19.86603126783482052510790435040, −19.34943469921956804720041544077, −17.97946977876735876359879568592, −17.63309445802167493877928182698, −16.69698316354182553028375128871, −15.68860421097890440650397960088, −14.60262980251217101383151567897, −13.62622138338885122415271031130, −12.85314299301711086902929743841, −11.78637596020089289342888377324, −10.47031012213558119484861999756, −9.622333838952073586497364503673, −8.79904020808582142756762498041, −7.99925974559571883021842251933, −6.919976317823251782775082260485, −6.301448806944058597963984750489, −4.62986243795363468576491751588, −3.139376345174649211886245409566, −1.717575289638626551223526836025, −1.14230422843319905335616244964, 1.606227662076984412617987681834, 2.95312122181023587553492757481, 3.20942570599775996790726465338, 5.34608163241824963125412047458, 6.26797832029696309312086790549, 7.5148188475782257369403721562, 8.656255504399787507746966397200, 9.14776698127775562692022675568, 10.11114136577145282826863875814, 10.93862363138083646929741280228, 11.76271873718535829858696411229, 13.341823071284737849457285271434, 14.18035745848571659841611632660, 15.32121347637461532071465570007, 15.82751466697597012012922254813, 16.79644698603791467915788076033, 18.05775534086177385733114768520, 18.63745650825559357592602976437, 19.311336877119179295453648420134, 20.51092692645964080661400558778, 21.15780955184446890662184565181, 21.85816499530861244358288285428, 22.7177966588951445744766285959, 24.587699190294259115523370885392, 25.04076047328137554915391530882

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