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Properties

Label 1-379-379.17-r1-0-0
Degree $1$
Conductor $379$
Sign $-0.196 + 0.980i$
Analytic cond. $40.7292$
Root an. cond. $40.7292$
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  + (−0.334 + 0.942i)2-s + (−0.426 − 0.904i)3-s + (−0.776 − 0.629i)4-s + (0.365 − 0.930i)5-s + (0.995 − 0.0995i)6-s + (0.556 + 0.830i)7-s + (0.853 − 0.521i)8-s + (−0.636 + 0.771i)9-s + (0.755 + 0.655i)10-s + (0.993 + 0.116i)11-s + (−0.238 + 0.971i)12-s + (0.157 + 0.987i)13-s + (−0.969 + 0.246i)14-s + (−0.997 + 0.0664i)15-s + (0.206 + 0.978i)16-s + (0.996 − 0.0830i)17-s + ⋯
L(s)  = 1  + (−0.334 + 0.942i)2-s + (−0.426 − 0.904i)3-s + (−0.776 − 0.629i)4-s + (0.365 − 0.930i)5-s + (0.995 − 0.0995i)6-s + (0.556 + 0.830i)7-s + (0.853 − 0.521i)8-s + (−0.636 + 0.771i)9-s + (0.755 + 0.655i)10-s + (0.993 + 0.116i)11-s + (−0.238 + 0.971i)12-s + (0.157 + 0.987i)13-s + (−0.969 + 0.246i)14-s + (−0.997 + 0.0664i)15-s + (0.206 + 0.978i)16-s + (0.996 − 0.0830i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 379 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(379\)
Sign: $-0.196 + 0.980i$
Analytic conductor: \(40.7292\)
Root analytic conductor: \(40.7292\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{379} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 379,\ (1:\ ),\ -0.196 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6800347605 + 0.8302158711i\)
\(L(\frac12)\) \(\approx\) \(0.6800347605 + 0.8302158711i\)
\(L(1)\) \(\approx\) \(0.7900357425 + 0.1928670065i\)
\(L(1)\) \(\approx\) \(0.7900357425 + 0.1928670065i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad379 \( 1 \)
good2 \( 1 + (-0.334 + 0.942i)T \)
3 \( 1 + (-0.426 - 0.904i)T \)
5 \( 1 + (0.365 - 0.930i)T \)
7 \( 1 + (0.556 + 0.830i)T \)
11 \( 1 + (0.993 + 0.116i)T \)
13 \( 1 + (0.157 + 0.987i)T \)
17 \( 1 + (0.996 - 0.0830i)T \)
19 \( 1 + (-0.870 - 0.492i)T \)
23 \( 1 + (-0.318 + 0.947i)T \)
29 \( 1 + (-0.878 - 0.478i)T \)
31 \( 1 + (-0.674 + 0.738i)T \)
37 \( 1 + (-0.0249 - 0.999i)T \)
41 \( 1 + (-0.0249 + 0.999i)T \)
43 \( 1 + (-0.945 - 0.326i)T \)
47 \( 1 + (-0.107 + 0.994i)T \)
53 \( 1 + (-0.0415 + 0.999i)T \)
59 \( 1 + (0.661 + 0.749i)T \)
61 \( 1 + (-0.0581 - 0.998i)T \)
67 \( 1 + (-0.411 + 0.911i)T \)
71 \( 1 + (-0.206 + 0.978i)T \)
73 \( 1 + (0.583 + 0.811i)T \)
79 \( 1 + (-0.835 + 0.549i)T \)
83 \( 1 + (0.542 + 0.840i)T \)
89 \( 1 + (0.349 + 0.936i)T \)
97 \( 1 + (0.00831 - 0.999i)T \)
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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

−23.722864114612389059016713034928, −22.68231359462350825774898124778, −22.39992091705207325134568555774, −21.38782472629499041653816321887, −20.67086566448873754407190580766, −19.9566329730914700728001224986, −18.77410041107009842477763985180, −17.94609493336683475029345214281, −17.05573327838889800126121017478, −16.62126911983853977217957728135, −14.805070279792434501489611948164, −14.48757887595217961661406391964, −13.282451070021798841776159985698, −12.01360485068183133839711845619, −11.1812796088472138346032106631, −10.38898040421184129412536929090, −10.0328936820237732996208034881, −8.78557105008936178756117127654, −7.691544796570021651628312302069, −6.32952544748812332228997607998, −5.12335932006830489559610026829, −3.86148896157530062608248110178, −3.36487900808448641290009763096, −1.79912323688744298619869942003, −0.391281968474349625983856981178, 1.20672492760140026875642767996, 1.88178926038696355977801053080, 4.25444157762330257523834802511, 5.35062097313697573070225128841, 5.97207486647238036301192856010, 6.98194161967253615788883017187, 8.0506388130936568754756399461, 8.87276662157552945806982047598, 9.56394917094798747557601766738, 11.22300402570644660301459384234, 12.113762378746580631274403898048, 13.00001777009155771927133097047, 14.00557148651660108360728110068, 14.68766330495203238589047602857, 16.00612327642060937563151945468, 16.85940411263792863300799806116, 17.37319360974684408555895140444, 18.25170999980121522281792428770, 19.08239339428319744491155650505, 19.84454921110302253277777684377, 21.34720637197730931319911166047, 22.05662792154211954403137715052, 23.365111974332455498553093942372, 23.83205273560417630425419987001, 24.71681013018006698481374910144

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