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Properties

Label 2-555-185.184-c1-0-19
Degree $2$
Conductor $555$
Sign $0.625 + 0.780i$
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  − 1.72·2-s + i·3-s + 0.980·4-s + (−1.26 + 1.84i)5-s − 1.72i·6-s − 2.83i·7-s + 1.75·8-s − 9-s + (2.17 − 3.18i)10-s + 3.10·11-s + 0.980i·12-s − 5.74·13-s + 4.88i·14-s + (−1.84 − 1.26i)15-s − 4.99·16-s − 4.52·17-s + ⋯
L(s)  = 1  − 1.22·2-s + 0.577i·3-s + 0.490·4-s + (−0.563 + 0.826i)5-s − 0.704i·6-s − 1.07i·7-s + 0.622·8-s − 0.333·9-s + (0.687 − 1.00i)10-s + 0.935·11-s + 0.283i·12-s − 1.59·13-s + 1.30i·14-s + (−0.476 − 0.325i)15-s − 1.24·16-s − 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(555\)    =    \(3 \cdot 5 \cdot 37\)
Sign: $0.625 + 0.780i$
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: \((2,\ 555,\ (\ :1/2),\ 0.625 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.401346 - 0.192610i\)
\(L(\frac12)\) \(\approx\) \(0.401346 - 0.192610i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.26 - 1.84i)T \)
37 \( 1 + (-5.81 - 1.77i)T \)
good2 \( 1 + 1.72T + 2T^{2} \)
7 \( 1 + 2.83iT - 7T^{2} \)
11 \( 1 - 3.10T + 11T^{2} \)
13 \( 1 + 5.74T + 13T^{2} \)
17 \( 1 + 4.52T + 17T^{2} \)
19 \( 1 + 1.75iT - 19T^{2} \)
23 \( 1 - 7.13T + 23T^{2} \)
29 \( 1 + 5.33iT - 29T^{2} \)
31 \( 1 + 4.68iT - 31T^{2} \)
41 \( 1 - 1.05T + 41T^{2} \)
43 \( 1 - 7.15T + 43T^{2} \)
47 \( 1 + 9.66iT - 47T^{2} \)
53 \( 1 - 7.77iT - 53T^{2} \)
59 \( 1 - 1.06iT - 59T^{2} \)
61 \( 1 + 8.12iT - 61T^{2} \)
67 \( 1 + 12.3iT - 67T^{2} \)
71 \( 1 - 3.54T + 71T^{2} \)
73 \( 1 + 5.95iT - 73T^{2} \)
79 \( 1 + 4.11iT - 79T^{2} \)
83 \( 1 - 10.8iT - 83T^{2} \)
89 \( 1 + 9.50iT - 89T^{2} \)
97 \( 1 - 2.02T + 97T^{2} \)
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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

−10.61504711875097024091378773135, −9.642550582884015990162272357402, −9.195966976202865774127343482175, −7.954303121595561872151861899907, −7.25141840886086455980608787854, −6.61220556684199318384426874950, −4.67689060935184396083235244656, −4.03722355144642129639526395841, −2.50266370783555982937593890339, −0.44510889879997592558987595949, 1.19468984875382362119501704597, 2.52099830006475735388761860745, 4.40618642891628747581723991194, 5.33517401139091934658960076467, 6.80757285864413558265963488667, 7.49130437936405982256564430823, 8.516371595723541877578559863154, 9.036003775727161199402942511104, 9.554853884345313799886395884192, 10.93292994502467694932277056662

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