Properties

Label 2-2775-111.11-c0-0-0
Degree $2$
Conductor $2775$
Sign $-0.227 + 0.973i$
Analytic cond. $1.38490$
Root an. cond. $1.17682$
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  + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s + (1.5 + 0.866i)13-s + (−0.499 − 0.866i)16-s + (−0.499 − 0.866i)21-s − 0.999·27-s + (−0.499 − 0.866i)28-s + 1.73i·31-s − 0.999·36-s − 37-s + (1.5 − 0.866i)39-s + 1.73i·43-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (0.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s + (1.5 + 0.866i)13-s + (−0.499 − 0.866i)16-s + (−0.499 − 0.866i)21-s − 0.999·27-s + (−0.499 − 0.866i)28-s + 1.73i·31-s − 0.999·36-s − 37-s + (1.5 − 0.866i)39-s + 1.73i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.227 + 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2775\)    =    \(3 \cdot 5^{2} \cdot 37\)
Sign: $-0.227 + 0.973i$
Analytic conductor: \(1.38490\)
Root analytic conductor: \(1.17682\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2775} (1676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2775,\ (\ :0),\ -0.227 + 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.776730985\)
\(L(\frac12)\) \(\approx\) \(1.776730985\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
5 \( 1 \)
37 \( 1 + T \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 1.73iT - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 - 1.73iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - 1.73iT - T^{2} \)
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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

−8.737746830402323761778979237602, −7.971484709414468762173790282772, −7.12322501217349445116362600144, −6.60572401394442785051130352317, −5.98000098094200671993789129484, −4.92486245114663513451730362960, −3.91341765843496044833779666399, −2.94763947132601989060785544738, −1.64491124363510222152541210383, −1.23657015470512178342727386420, 1.91908160474091000860488401732, 2.78150680868566652829432988207, 3.59762279093051355611247593360, 4.24079695779039540127202302746, 5.44634112571062042917450517953, 5.92538701598272688267222747163, 7.11804143952243880207139070366, 7.978756701008942807421859585060, 8.558061667358138619338182477038, 8.852073757962957248510621342561

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