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Properties

Label 1-373-373.3-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.725 - 0.687i$
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  + (−0.638 − 0.769i)2-s + (−0.117 + 0.993i)3-s + (−0.184 + 0.982i)4-s + (0.713 + 0.701i)5-s + (0.839 − 0.543i)6-s + (−0.440 − 0.897i)7-s + (0.874 − 0.485i)8-s + (−0.972 − 0.234i)9-s + (0.0843 − 0.996i)10-s + (−0.0168 − 0.999i)11-s + (−0.954 − 0.299i)12-s + (−0.874 − 0.485i)13-s + (−0.409 + 0.912i)14-s + (−0.780 + 0.625i)15-s + (−0.931 − 0.363i)16-s + (−0.758 − 0.651i)17-s + ⋯
L(s)  = 1  + (−0.638 − 0.769i)2-s + (−0.117 + 0.993i)3-s + (−0.184 + 0.982i)4-s + (0.713 + 0.701i)5-s + (0.839 − 0.543i)6-s + (−0.440 − 0.897i)7-s + (0.874 − 0.485i)8-s + (−0.972 − 0.234i)9-s + (0.0843 − 0.996i)10-s + (−0.0168 − 0.999i)11-s + (−0.954 − 0.299i)12-s + (−0.874 − 0.485i)13-s + (−0.409 + 0.912i)14-s + (−0.780 + 0.625i)15-s + (−0.931 − 0.363i)16-s + (−0.758 − 0.651i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.725 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.725 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.725 - 0.687i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.725 - 0.687i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1130343928 - 0.2836618051i\)
\(L(\frac12)\) \(\approx\) \(0.1130343928 - 0.2836618051i\)
\(L(1)\) \(\approx\) \(0.5649911760 - 0.08889665713i\)
\(L(1)\) \(\approx\) \(0.5649911760 - 0.08889665713i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.638 - 0.769i)T \)
3 \( 1 + (-0.117 + 0.993i)T \)
5 \( 1 + (0.713 + 0.701i)T \)
7 \( 1 + (-0.440 - 0.897i)T \)
11 \( 1 + (-0.0168 - 0.999i)T \)
13 \( 1 + (-0.874 - 0.485i)T \)
17 \( 1 + (-0.758 - 0.651i)T \)
19 \( 1 + (-0.151 + 0.988i)T \)
23 \( 1 + (-0.918 - 0.394i)T \)
29 \( 1 + (-0.315 - 0.948i)T \)
31 \( 1 + (-0.954 + 0.299i)T \)
37 \( 1 + (-0.999 - 0.0337i)T \)
41 \( 1 + (-0.0506 - 0.998i)T \)
43 \( 1 + (0.184 - 0.982i)T \)
47 \( 1 + (-0.585 + 0.810i)T \)
53 \( 1 + (0.972 - 0.234i)T \)
59 \( 1 + (-0.664 + 0.747i)T \)
61 \( 1 + (0.117 - 0.993i)T \)
67 \( 1 + (0.250 - 0.968i)T \)
71 \( 1 + (0.943 + 0.331i)T \)
73 \( 1 + (0.283 - 0.959i)T \)
79 \( 1 + (-0.0843 + 0.996i)T \)
83 \( 1 + (-0.972 + 0.234i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.758 + 0.651i)T \)
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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

−24.83987749359972130615977988281, −24.33273818181478989308201330783, −23.64072432193106675566940960687, −22.4485715144232617235527920689, −21.67642477695257231055609175872, −19.92516393356237177669178118296, −19.74914436237634663056251460183, −18.46603914620056300437465143523, −17.83329316422814052592828393848, −17.19937147483073899688926247005, −16.30951545849276734120514311112, −15.19034815602910046572313432083, −14.34073293171614032148631915827, −13.19481900163181659798956987090, −12.59662001219608234068722039314, −11.48583772856248727259879144796, −10.01728736796718879419406465787, −9.168101400854628171311650731710, −8.51075156673204680927967997521, −7.268219260300530890740368928737, −6.51515937475515156947196994535, −5.59723769824005519983128360176, −4.74267680876901380880807986719, −2.28365672577401593819414964730, −1.69054412192240946620360466581, 0.21282829374441143868594210027, 2.20916552163473791724238561022, 3.26493933945820164884412538528, 4.021502733684610664237985975, 5.4539708802558368590737530948, 6.689056971128073628388608992240, 7.87754457823699599107067249314, 9.09840457661449649308321296332, 9.92127994816594582054912317560, 10.51277910148161677552944729863, 11.149693595141230793261938068112, 12.35178461909545687634492545078, 13.66976063656362616186338904744, 14.19801030085761080508567934599, 15.61683729567258800904246028732, 16.61271702221059924478451223609, 17.16637553085723695944403566568, 18.07357300453236978763253725825, 19.111312688900250111249527124506, 20.01022605323571084725946183335, 20.76493835574588396896447673694, 21.58721669316075981541830863147, 22.44639984204296748643523407137, 22.73049049108514913717437172653, 24.453918686206759069908048300878

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