Properties

Label 1-373-373.140-r1-0-0
Degree $1$
Conductor $373$
Sign $0.762 + 0.646i$
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  + (0.998 − 0.0506i)2-s + (0.874 + 0.485i)3-s + (0.994 − 0.101i)4-s + (−0.651 − 0.758i)5-s + (0.897 + 0.440i)6-s + (−0.820 + 0.571i)7-s + (0.988 − 0.151i)8-s + (0.528 + 0.848i)9-s + (−0.688 − 0.724i)10-s + (0.988 + 0.151i)11-s + (0.918 + 0.394i)12-s + (−0.151 + 0.988i)13-s + (−0.790 + 0.612i)14-s + (−0.201 − 0.979i)15-s + (0.979 − 0.201i)16-s + (0.994 − 0.101i)17-s + ⋯
L(s)  = 1  + (0.998 − 0.0506i)2-s + (0.874 + 0.485i)3-s + (0.994 − 0.101i)4-s + (−0.651 − 0.758i)5-s + (0.897 + 0.440i)6-s + (−0.820 + 0.571i)7-s + (0.988 − 0.151i)8-s + (0.528 + 0.848i)9-s + (−0.688 − 0.724i)10-s + (0.988 + 0.151i)11-s + (0.918 + 0.394i)12-s + (−0.151 + 0.988i)13-s + (−0.790 + 0.612i)14-s + (−0.201 − 0.979i)15-s + (0.979 − 0.201i)16-s + (0.994 − 0.101i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $0.762 + 0.646i$
Analytic conductor: \(40.0844\)
Root analytic conductor: \(40.0844\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (1:\ ),\ 0.762 + 0.646i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.680069577 + 1.716662170i\)
\(L(\frac12)\) \(\approx\) \(4.680069577 + 1.716662170i\)
\(L(1)\) \(\approx\) \(2.454998522 + 0.4101422949i\)
\(L(1)\) \(\approx\) \(2.454998522 + 0.4101422949i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (0.998 - 0.0506i)T \)
3 \( 1 + (0.874 + 0.485i)T \)
5 \( 1 + (-0.651 - 0.758i)T \)
7 \( 1 + (-0.820 + 0.571i)T \)
11 \( 1 + (0.988 + 0.151i)T \)
13 \( 1 + (-0.151 + 0.988i)T \)
17 \( 1 + (0.994 - 0.101i)T \)
19 \( 1 + (0.201 - 0.979i)T \)
23 \( 1 + (0.485 + 0.874i)T \)
29 \( 1 + (-0.250 - 0.968i)T \)
31 \( 1 + (-0.918 + 0.394i)T \)
37 \( 1 + (0.954 - 0.299i)T \)
41 \( 1 + (-0.440 + 0.897i)T \)
43 \( 1 + (0.101 + 0.994i)T \)
47 \( 1 + (0.790 + 0.612i)T \)
53 \( 1 + (0.848 + 0.528i)T \)
59 \( 1 + (0.250 + 0.968i)T \)
61 \( 1 + (0.485 - 0.874i)T \)
67 \( 1 + (-0.651 - 0.758i)T \)
71 \( 1 + (0.994 + 0.101i)T \)
73 \( 1 + (0.528 - 0.848i)T \)
79 \( 1 + (-0.724 + 0.688i)T \)
83 \( 1 + (0.528 - 0.848i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.101 - 0.994i)T \)
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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

−24.175216490542011393479023009775, −23.34879621683163781676434722747, −22.6410332106588553090711237585, −21.954353026505185577573653160763, −20.51851719782946413723597756054, −20.09254963531709974357642770108, −19.22091280937862230949545083943, −18.54123281827996535738432777497, −16.93403831620541960628716484827, −16.08280142631800860432926666575, −14.90708682613874857320280051417, −14.564293550037776672707840774927, −13.65212882613578565694331061474, −12.62157865751085619666098268497, −12.07914151382796056966043434593, −10.74112620956789904279967396438, −9.89265112774899916336839716447, −8.34111711457822013200537751662, −7.33171281041651543113957316150, −6.82283798313517570912135593830, −5.71760444293937524842677006273, −3.81726157238450676471636131285, −3.58562583723195699283380053104, −2.54559918758439361873003675921, −0.976193213118569894226191600012, 1.40719644466162891869291675141, 2.760847416773699538730627682457, 3.71547062396296043500511834352, 4.448056222551536780045015004327, 5.51829281375893881845196799890, 6.85807957259167492473653906275, 7.78921382052440484516209691528, 9.16274848816547908228063177628, 9.59212382027959268848989513680, 11.23888088745529878302324297138, 12.008143484463187511110433905975, 12.91283074563620602685447370811, 13.73441806887597772903795419029, 14.7612795003657953372579608929, 15.40545665926926238473930103028, 16.30101832934680249276002278921, 16.80350976334259815316599597092, 18.90917451060699175558540864650, 19.61169365981171665135138031353, 20.02889536033665789988633359381, 21.2260209680391688626591183578, 21.68847652255284524363862306620, 22.67929675313248087430161366195, 23.60523411787611218501560997771, 24.553555538068306395220222600294

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