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Properties

Label 2-184-184.91-c1-0-5
Degree $2$
Conductor $184$
Sign $0.109 - 0.994i$
Analytic cond. $1.46924$
Root an. cond. $1.21212$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)2-s − 2·3-s + 2i·4-s + 3.74·5-s + (−2 − 2i)6-s + (−2 + 2i)8-s + 9-s + (3.74 + 3.74i)10-s + 3.74i·11-s − 4i·12-s + 4i·13-s − 7.48·15-s − 4·16-s − 7.48i·17-s + (1 + i)18-s − 3.74i·19-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s − 1.15·3-s + i·4-s + 1.67·5-s + (−0.816 − 0.816i)6-s + (−0.707 + 0.707i)8-s + 0.333·9-s + (1.18 + 1.18i)10-s + 1.12i·11-s − 1.15i·12-s + 1.10i·13-s − 1.93·15-s − 16-s − 1.81i·17-s + (0.235 + 0.235i)18-s − 0.858i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.109 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.109 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(184\)    =    \(2^{3} \cdot 23\)
Sign: $0.109 - 0.994i$
Analytic conductor: \(1.46924\)
Root analytic conductor: \(1.21212\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{184} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 184,\ (\ :1/2),\ 0.109 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05610 + 0.946295i\)
\(L(\frac12)\) \(\approx\) \(1.05610 + 0.946295i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
23 \( 1 + (-3.74 + 3i)T \)
good3 \( 1 + 2T + 3T^{2} \)
5 \( 1 - 3.74T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 3.74iT - 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + 7.48iT - 17T^{2} \)
19 \( 1 + 3.74iT - 19T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 2iT - 31T^{2} \)
37 \( 1 + 3.74T + 37T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + 3.74iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 + 3.74T + 53T^{2} \)
59 \( 1 + 2T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 + 11.2iT - 67T^{2} \)
71 \( 1 - 12iT - 71T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 + 7.48T + 79T^{2} \)
83 \( 1 - 11.2iT - 83T^{2} \)
89 \( 1 - 7.48iT - 89T^{2} \)
97 \( 1 + 7.48iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.95994284175456130221658717946, −11.99940431993657113084740908344, −11.15107403426225027003690027295, −9.783050857490747431568234566074, −8.989207658483847820091447243225, −6.93300420890005330966792667750, −6.60643130094261070477138724405, −5.25953054986508603221972794454, −4.83031192695072343460184636443, −2.42136480627944769424890643112, 1.43921759295853468101761526996, 3.23924115048015844390918395220, 5.17701535225859987080360240440, 5.85371545182580844712772491264, 6.32854345382886087974174240767, 8.608616022707566008882485854245, 9.997098492649248403394385694537, 10.55816539214557612817551038674, 11.28560761110908356557053600897, 12.59049661668780874930029697896

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