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Properties

Label 1-373-373.95-r0-0-0
Degree $1$
Conductor $373$
Sign $-0.636 + 0.771i$
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  + (−0.985 − 0.168i)2-s + (−0.801 + 0.598i)3-s + (0.943 + 0.331i)4-s + (0.963 + 0.266i)5-s + (0.890 − 0.455i)6-s + (−0.440 + 0.897i)7-s + (−0.874 − 0.485i)8-s + (0.283 − 0.959i)9-s + (−0.905 − 0.425i)10-s + (0.857 + 0.514i)11-s + (−0.954 + 0.299i)12-s + (−0.874 + 0.485i)13-s + (0.585 − 0.810i)14-s + (−0.931 + 0.363i)15-s + (0.780 + 0.625i)16-s + (−0.758 + 0.651i)17-s + ⋯
L(s)  = 1  + (−0.985 − 0.168i)2-s + (−0.801 + 0.598i)3-s + (0.943 + 0.331i)4-s + (0.963 + 0.266i)5-s + (0.890 − 0.455i)6-s + (−0.440 + 0.897i)7-s + (−0.874 − 0.485i)8-s + (0.283 − 0.959i)9-s + (−0.905 − 0.425i)10-s + (0.857 + 0.514i)11-s + (−0.954 + 0.299i)12-s + (−0.874 + 0.485i)13-s + (0.585 − 0.810i)14-s + (−0.931 + 0.363i)15-s + (0.780 + 0.625i)16-s + (−0.758 + 0.651i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 373 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(373\)
Sign: $-0.636 + 0.771i$
Analytic conductor: \(1.73220\)
Root analytic conductor: \(1.73220\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{373} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 373,\ (0:\ ),\ -0.636 + 0.771i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2479568520 + 0.5263418991i\)
\(L(\frac12)\) \(\approx\) \(0.2479568520 + 0.5263418991i\)
\(L(1)\) \(\approx\) \(0.5359395884 + 0.2556696646i\)
\(L(1)\) \(\approx\) \(0.5359395884 + 0.2556696646i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad373 \( 1 \)
good2 \( 1 + (-0.985 - 0.168i)T \)
3 \( 1 + (-0.801 + 0.598i)T \)
5 \( 1 + (0.963 + 0.266i)T \)
7 \( 1 + (-0.440 + 0.897i)T \)
11 \( 1 + (0.857 + 0.514i)T \)
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 \( 1 + (-0.664 - 0.747i)T \)
31 \( 1 + (-0.954 - 0.299i)T \)
37 \( 1 + (0.470 - 0.882i)T \)
41 \( 1 + (-0.0506 + 0.998i)T \)
43 \( 1 + (0.943 + 0.331i)T \)
47 \( 1 + (0.409 - 0.912i)T \)
53 \( 1 + (0.283 + 0.959i)T \)
59 \( 1 + (-0.315 + 0.948i)T \)
61 \( 1 + (-0.801 + 0.598i)T \)
67 \( 1 + (-0.250 - 0.968i)T \)
71 \( 1 + (-0.184 + 0.982i)T \)
73 \( 1 + (-0.972 - 0.234i)T \)
79 \( 1 + (-0.905 - 0.425i)T \)
83 \( 1 + (0.283 + 0.959i)T \)
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

−24.379412247278655002185833848925, −23.821358860606557703741268632778, −22.44663704308213471576813374467, −21.87019873375064111333551937363, −20.46520791718677741757082814923, −19.75672206920658240004629541943, −18.91525170338481864130218979777, −17.8215871339767057686514971236, −17.27313484507123465999475369397, −16.76715000904579705736302987050, −15.86391613657145191378861794346, −14.41035735229715997655881023658, −13.41803230522204531225461223811, −12.56617080372113209331372167136, −11.31399489091584924606926756444, −10.68701474976732778104080435225, −9.62436203406543632359987319433, −8.897794971463737982932896626431, −7.30936407897036636756947948636, −6.90117706348775546764514020191, −5.88416649615212379408916248053, −4.89158103995354824720688842513, −2.865066086238947358145738818972, −1.57109598059805565222196474097, −0.54515789418581819719362528877, 1.59237417153691275674013683241, 2.63193281475576390940813601069, 4.110574256286872839006693425895, 5.66347503776278145277525076501, 6.30960503610792920531276355982, 7.2179721427982680157250117401, 9.03870957243020348958728066559, 9.38568136572747848534475921272, 10.227105309770649710526236268707, 11.17094097860926483542054740087, 12.11721297492000799504536632066, 12.82838392993807209149785528422, 14.72139177235705868714565165812, 15.16801162577398992822599497580, 16.56064582061534539020521226278, 16.92592841593014507657638778459, 17.84336698459667858933068413997, 18.54143489476132418923463273772, 19.51372679333574707739708408464, 20.634928819726748156845590225422, 21.57140942944164969508082187801, 22.00799282223521535187441159528, 22.89253840762373210460400782947, 24.50732383071579891560474472381, 24.94751597027051000370787994562

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