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Properties

Label 2-555-185.184-c1-0-38
Degree $2$
Conductor $555$
Sign $0.421 + 0.906i$
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  + 2.49·2-s − i·3-s + 4.23·4-s + (−2.05 − 0.879i)5-s − 2.49i·6-s − 3.27i·7-s + 5.57·8-s − 9-s + (−5.13 − 2.19i)10-s − 2.24·11-s − 4.23i·12-s + 3.94·13-s − 8.18i·14-s + (−0.879 + 2.05i)15-s + 5.45·16-s + 5.46·17-s + ⋯
L(s)  = 1  + 1.76·2-s − 0.577i·3-s + 2.11·4-s + (−0.919 − 0.393i)5-s − 1.01i·6-s − 1.23i·7-s + 1.97·8-s − 0.333·9-s + (−1.62 − 0.694i)10-s − 0.678·11-s − 1.22i·12-s + 1.09·13-s − 2.18i·14-s + (−0.227 + 0.530i)15-s + 1.36·16-s + 1.32·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.421 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.421 + 0.906i$
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: \((2,\ 555,\ (\ :1/2),\ 0.421 + 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.90540 - 1.85236i\)
\(L(\frac12)\) \(\approx\) \(2.90540 - 1.85236i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.05 + 0.879i)T \)
37 \( 1 + (4.05 - 4.52i)T \)
good2 \( 1 - 2.49T + 2T^{2} \)
7 \( 1 + 3.27iT - 7T^{2} \)
11 \( 1 + 2.24T + 11T^{2} \)
13 \( 1 - 3.94T + 13T^{2} \)
17 \( 1 - 5.46T + 17T^{2} \)
19 \( 1 - 4.91iT - 19T^{2} \)
23 \( 1 + 0.293T + 23T^{2} \)
29 \( 1 + 4.42iT - 29T^{2} \)
31 \( 1 - 9.50iT - 31T^{2} \)
41 \( 1 + 7.53T + 41T^{2} \)
43 \( 1 - 9.74T + 43T^{2} \)
47 \( 1 - 5.70iT - 47T^{2} \)
53 \( 1 + 14.1iT - 53T^{2} \)
59 \( 1 - 4.15iT - 59T^{2} \)
61 \( 1 + 1.76iT - 61T^{2} \)
67 \( 1 - 2.23iT - 67T^{2} \)
71 \( 1 + 1.23T + 71T^{2} \)
73 \( 1 + 2.78iT - 73T^{2} \)
79 \( 1 + 7.60iT - 79T^{2} \)
83 \( 1 + 7.35iT - 83T^{2} \)
89 \( 1 - 11.7iT - 89T^{2} \)
97 \( 1 + 8.37T + 97T^{2} \)
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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

−10.95016330353357011751526900631, −10.24935099864660780968696339150, −8.350110713375400001384462768969, −7.65148716552791941882613254194, −6.86729200168408025834301628813, −5.83171934187086887365423506657, −4.89093259750531347347621744150, −3.80843975707248862319127101805, −3.27229689714192271553524378852, −1.33259208844249981272682789599, 2.55719645750242473219077189745, 3.33961584889381287585950453921, 4.24408249198200554885333936900, 5.34851469391333156071168696712, 5.87258404930228377148015019174, 7.05365054661954221248379321804, 8.103022546610349222040813711402, 9.139761826435297065096612831034, 10.59525517198068234316378155916, 11.19398226874779405840701232144

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