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Properties

Label 2-2268-63.41-c1-0-13
Degree $2$
Conductor $2268$
Sign $0.356 + 0.934i$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
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  + (−1.5 + 2.59i)5-s + (−2.5 − 0.866i)7-s + (−4.5 + 2.59i)11-s + (3 + 1.73i)13-s − 6·17-s − 1.73i·19-s + (−4.5 − 2.59i)23-s + (−2 − 3.46i)25-s + (9 − 5.19i)29-s + (4.5 + 2.59i)31-s + (6 − 5.19i)35-s + 37-s + (1.5 − 2.59i)41-s + (−5 − 8.66i)43-s + (3 + 5.19i)47-s + ⋯
L(s)  = 1  + (−0.670 + 1.16i)5-s + (−0.944 − 0.327i)7-s + (−1.35 + 0.783i)11-s + (0.832 + 0.480i)13-s − 1.45·17-s − 0.397i·19-s + (−0.938 − 0.541i)23-s + (−0.400 − 0.692i)25-s + (1.67 − 0.964i)29-s + (0.808 + 0.466i)31-s + (1.01 − 0.878i)35-s + 0.164·37-s + (0.234 − 0.405i)41-s + (−0.762 − 1.32i)43-s + (0.437 + 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.356 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.356 + 0.934i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.356 + 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4881933945\)
\(L(\frac12)\) \(\approx\) \(0.4881933945\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.5 - 2.59i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3 - 1.73i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 + (4.5 + 2.59i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-9 + 5.19i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.5 - 2.59i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5 + 8.66i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (3 - 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-12 + 6.92i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.19iT - 71T^{2} \)
73 \( 1 - 3.46iT - 73T^{2} \)
79 \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3 + 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (-6 + 3.46i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.754164703075188022570699082981, −8.069146415829963395603865853482, −7.11106203635024905451752668965, −6.72620097489289444206399964457, −6.00745279469892424783902403835, −4.61882487654570121696261685868, −4.00609325707489503579030891712, −2.92345258787829591453167889765, −2.31849614996403011751768053581, −0.21245678210612668012323254067, 0.894959951490806760427160439264, 2.52621252122188378306326033244, 3.42486199155014047932433868534, 4.36452971064340424430364140235, 5.20656565635366265092621270037, 5.99582985030100934818224498391, 6.73047747157985758922350490261, 8.088736866739174323397495007089, 8.257543416871809701758181513378, 8.965582134882019551639942972133

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