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Properties

Label 2-555-185.184-c1-0-26
Degree $2$
Conductor $555$
Sign $-0.296 + 0.955i$
Analytic cond. $4.43169$
Root an. cond. $2.10515$
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  − 0.414·2-s − i·3-s − 1.82·4-s + (1.55 − 1.60i)5-s + 0.414i·6-s − 0.262i·7-s + 1.58·8-s − 9-s + (−0.643 + 0.666i)10-s + 5.76·11-s + 1.82i·12-s − 2.13·13-s + 0.108i·14-s + (−1.60 − 1.55i)15-s + 2.99·16-s − 0.318·17-s + ⋯
L(s)  = 1  − 0.293·2-s − 0.577i·3-s − 0.914·4-s + (0.694 − 0.719i)5-s + 0.169i·6-s − 0.0990i·7-s + 0.561·8-s − 0.333·9-s + (−0.203 + 0.210i)10-s + 1.73·11-s + 0.527i·12-s − 0.593·13-s + 0.0290i·14-s + (−0.415 − 0.401i)15-s + 0.749·16-s − 0.0773·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.296 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $-0.296 + 0.955i$
Analytic conductor: \(4.43169\)
Root analytic conductor: \(2.10515\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{555} (184, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 555,\ (\ :1/2),\ -0.296 + 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.631174 - 0.856458i\)
\(L(\frac12)\) \(\approx\) \(0.631174 - 0.856458i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (-1.55 + 1.60i)T \)
37 \( 1 + (-2.74 - 5.43i)T \)
good2 \( 1 + 0.414T + 2T^{2} \)
7 \( 1 + 0.262iT - 7T^{2} \)
11 \( 1 - 5.76T + 11T^{2} \)
13 \( 1 + 2.13T + 13T^{2} \)
17 \( 1 + 0.318T + 17T^{2} \)
19 \( 1 + 6.26iT - 19T^{2} \)
23 \( 1 + 8.22T + 23T^{2} \)
29 \( 1 + 5.91iT - 29T^{2} \)
31 \( 1 + 1.32iT - 31T^{2} \)
41 \( 1 + 5.96T + 41T^{2} \)
43 \( 1 - 0.0737T + 43T^{2} \)
47 \( 1 + 8.04iT - 47T^{2} \)
53 \( 1 + 5.23iT - 53T^{2} \)
59 \( 1 - 7.31iT - 59T^{2} \)
61 \( 1 + 2.66iT - 61T^{2} \)
67 \( 1 - 3.69iT - 67T^{2} \)
71 \( 1 - 7.33T + 71T^{2} \)
73 \( 1 + 2.43iT - 73T^{2} \)
79 \( 1 + 10.9iT - 79T^{2} \)
83 \( 1 - 13.2iT - 83T^{2} \)
89 \( 1 - 9.15iT - 89T^{2} \)
97 \( 1 - 1.87T + 97T^{2} \)
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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

−10.14557304181802042471878057164, −9.512069860884317524544775318463, −8.838175896042741544688998484741, −8.078840844033349763588375499449, −6.87167877685464163168219182468, −5.96624704831560500178265637928, −4.83443162425303771086938343706, −3.95849648235990380687079045294, −2.04869108748078747392453771488, −0.73657751274816202219007191179, 1.72216290432678242059403647385, 3.49413736151333933876693938617, 4.26991037722778480278906649084, 5.55251762609258488568606276356, 6.35633099771102154375643998816, 7.56860543412986503009308501620, 8.675041393323291278645604792237, 9.453094727016472592524089364944, 9.963381852702533132881026651970, 10.71121847938259005234607537294

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