Properties

Label 1-837-837.761-r0-0-0
Degree $1$
Conductor $837$
Sign $0.181 - 0.983i$
Analytic cond. $3.88701$
Root an. cond. $3.88701$
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  + (0.719 − 0.694i)2-s + (0.0348 − 0.999i)4-s + (0.939 + 0.342i)5-s + (0.990 − 0.139i)7-s + (−0.669 − 0.743i)8-s + (0.913 − 0.406i)10-s + (0.990 − 0.139i)11-s + (0.241 − 0.970i)13-s + (0.615 − 0.788i)14-s + (−0.997 − 0.0697i)16-s + (0.309 − 0.951i)17-s + (−0.104 + 0.994i)19-s + (0.374 − 0.927i)20-s + (0.615 − 0.788i)22-s + (−0.615 + 0.788i)23-s + ⋯
L(s)  = 1  + (0.719 − 0.694i)2-s + (0.0348 − 0.999i)4-s + (0.939 + 0.342i)5-s + (0.990 − 0.139i)7-s + (−0.669 − 0.743i)8-s + (0.913 − 0.406i)10-s + (0.990 − 0.139i)11-s + (0.241 − 0.970i)13-s + (0.615 − 0.788i)14-s + (−0.997 − 0.0697i)16-s + (0.309 − 0.951i)17-s + (−0.104 + 0.994i)19-s + (0.374 − 0.927i)20-s + (0.615 − 0.788i)22-s + (−0.615 + 0.788i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.181 - 0.983i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (761, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.181 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.285022963 - 1.901170098i\)
\(L(\frac12)\) \(\approx\) \(2.285022963 - 1.901170098i\)
\(L(1)\) \(\approx\) \(1.773201648 - 0.9030389752i\)
\(L(1)\) \(\approx\) \(1.773201648 - 0.9030389752i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.719 - 0.694i)T \)
5 \( 1 + (0.939 + 0.342i)T \)
7 \( 1 + (0.990 - 0.139i)T \)
11 \( 1 + (0.990 - 0.139i)T \)
13 \( 1 + (0.241 - 0.970i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.104 + 0.994i)T \)
23 \( 1 + (-0.615 + 0.788i)T \)
29 \( 1 + (-0.719 + 0.694i)T \)
37 \( 1 - T \)
41 \( 1 + (0.997 - 0.0697i)T \)
43 \( 1 + (-0.961 - 0.275i)T \)
47 \( 1 + (-0.559 + 0.829i)T \)
53 \( 1 + (-0.978 + 0.207i)T \)
59 \( 1 + (0.719 + 0.694i)T \)
61 \( 1 + (0.939 - 0.342i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.848 + 0.529i)T \)
83 \( 1 + (0.961 + 0.275i)T \)
89 \( 1 + (-0.978 - 0.207i)T \)
97 \( 1 + (-0.374 + 0.927i)T \)
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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

−22.07053011525155807565933374343, −21.72857281864570459533816206943, −20.955024037130852215826690869816, −20.30407717062091466943077472195, −19.05095541749407631242488986196, −17.93578562431491174271422236317, −17.32989649875744933063812448794, −16.8120305511512921659726087277, −15.8913851236541355251643693640, −14.67812617300132610481387772594, −14.43382486137132244201318266895, −13.56015548764309177377406846459, −12.7580700658404170743053028290, −11.82869857615683203485826613065, −11.152746183929560727070231822181, −9.80481076205117893650585871775, −8.800351419647016216746143738464, −8.31731233545074514395401280022, −7.00200673977974805576376363067, −6.32217488141452867828287164193, −5.46587079253833041954512771008, −4.56469664897895003249442505138, −3.87854707937193595731222492094, −2.33439542759018139539168561663, −1.56577468219085777802129094949, 1.24108166310360565429144370153, 1.858267794063679703532532002554, 3.07164345671062108711542714303, 3.89870055069939771873198584499, 5.14180219745982930820129924377, 5.66111468909186965624355290604, 6.621474844723280788474770968607, 7.74086008616764885833328931686, 9.00749228917604606648197500620, 9.82219686043390576847548258225, 10.60422915112875510469525043105, 11.3570048346739555456078976579, 12.141150243530059634197240380846, 13.09673253384015415644865334285, 14.03826957269601286752381190685, 14.3341390074013108601810610326, 15.14626231521452149280961043617, 16.29988115631827662338351010721, 17.42297283629243836183418540491, 18.04623992665625246890343396130, 18.773874049402266600749165859943, 19.8139928231664638767215140783, 20.685177010537981365283162146944, 21.019781096954050855411277640905, 22.05525087369424878115055577299

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