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Properties

Label 1-379-379.50-r1-0-0
Degree $1$
Conductor $379$
Sign $0.205 - 0.978i$
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  + (0.0912 − 0.995i)2-s + (−0.0415 − 0.999i)3-s + (−0.983 − 0.181i)4-s + (0.826 + 0.563i)5-s + (−0.998 − 0.0498i)6-s + (−0.999 − 0.0332i)7-s + (−0.270 + 0.962i)8-s + (−0.996 + 0.0830i)9-s + (0.636 − 0.771i)10-s + (0.835 − 0.549i)11-s + (−0.140 + 0.990i)12-s + (−0.334 + 0.942i)13-s + (−0.124 + 0.992i)14-s + (0.528 − 0.848i)15-s + (0.933 + 0.357i)16-s + (0.886 − 0.463i)17-s + ⋯
L(s)  = 1  + (0.0912 − 0.995i)2-s + (−0.0415 − 0.999i)3-s + (−0.983 − 0.181i)4-s + (0.826 + 0.563i)5-s + (−0.998 − 0.0498i)6-s + (−0.999 − 0.0332i)7-s + (−0.270 + 0.962i)8-s + (−0.996 + 0.0830i)9-s + (0.636 − 0.771i)10-s + (0.835 − 0.549i)11-s + (−0.140 + 0.990i)12-s + (−0.334 + 0.942i)13-s + (−0.124 + 0.992i)14-s + (0.528 − 0.848i)15-s + (0.933 + 0.357i)16-s + (0.886 − 0.463i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 379 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(379\)
Sign: $0.205 - 0.978i$
Analytic conductor: \(40.7292\)
Root analytic conductor: \(40.7292\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{379} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 379,\ (1:\ ),\ 0.205 - 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.338288063 - 1.085906654i\)
\(L(\frac12)\) \(\approx\) \(1.338288063 - 1.085906654i\)
\(L(1)\) \(\approx\) \(0.8603446508 - 0.6235858803i\)
\(L(1)\) \(\approx\) \(0.8603446508 - 0.6235858803i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad379 \( 1 \)
good2 \( 1 + (0.0912 - 0.995i)T \)
3 \( 1 + (-0.0415 - 0.999i)T \)
5 \( 1 + (0.826 + 0.563i)T \)
7 \( 1 + (-0.999 - 0.0332i)T \)
11 \( 1 + (0.835 - 0.549i)T \)
13 \( 1 + (-0.334 + 0.942i)T \)
17 \( 1 + (0.886 - 0.463i)T \)
19 \( 1 + (-0.710 + 0.704i)T \)
23 \( 1 + (-0.583 + 0.811i)T \)
29 \( 1 + (0.969 - 0.246i)T \)
31 \( 1 + (-0.107 - 0.994i)T \)
37 \( 1 + (0.698 + 0.715i)T \)
41 \( 1 + (0.698 - 0.715i)T \)
43 \( 1 + (0.349 - 0.936i)T \)
47 \( 1 + (0.950 - 0.310i)T \)
53 \( 1 + (-0.238 + 0.971i)T \)
59 \( 1 + (0.411 + 0.911i)T \)
61 \( 1 + (-0.286 + 0.957i)T \)
67 \( 1 + (0.542 - 0.840i)T \)
71 \( 1 + (-0.933 + 0.357i)T \)
73 \( 1 + (-0.456 - 0.889i)T \)
79 \( 1 + (0.973 - 0.230i)T \)
83 \( 1 + (0.878 - 0.478i)T \)
89 \( 1 + (-0.426 + 0.904i)T \)
97 \( 1 + (0.964 - 0.262i)T \)
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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

−24.95374867856333082543288510574, −23.53682331525408929530455259174, −22.75593465117458532462458075594, −21.98773631951554775676812897849, −21.43040335562458531732280074316, −20.17885426103267209171392770021, −19.396395995155853781960369370085, −17.825611977031004291399620048332, −17.28628702436846121816862413994, −16.43946414064958791189919162378, −15.86854336317686682879181851605, −14.74365044013887480657785869902, −14.20144438416635002413271874913, −12.878589691092482555085766403710, −12.37518885243169291857039100115, −10.43697024599002856333023917224, −9.76523425582164779323014222454, −9.095089488494104060770942347945, −8.163783277231324541041147138987, −6.601574888329187492567668530196, −5.92522399951775232441149463106, −4.94751736798779000297244450526, −4.05348025473670449000697214268, −2.82792473472019092447724515621, −0.651907218668235508582351252566, 0.8563926873845853536788485632, 1.99439697560382023380715045129, 2.87913069662890612729007272050, 3.941725499690504769453639657160, 5.746654908772965265267620165080, 6.27599107796788377504782741733, 7.47311005897266362057444334804, 8.91160023511009524307182761357, 9.62416218261568929342546350442, 10.586687900707559447441822731405, 11.78693520637167186993997960303, 12.28713781629080603052129545874, 13.525183067092731773267445998160, 13.86163832499449662361138095353, 14.72721897446765101430366728199, 16.6566570936050323693563056040, 17.25004121282572612708648001430, 18.366081758991100369382850446831, 19.05782651574347058728837671438, 19.41520438037164461503394989365, 20.615488202719241276060730977959, 21.68182608915178072321131958199, 22.30287470851616901110099685161, 23.10779709628357177474155447468, 23.939014353136860442081851474661

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