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Properties

Label 2-2268-63.59-c1-0-27
Degree $2$
Conductor $2268$
Sign $-0.270 + 0.962i$
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  + 1.13·5-s + (−1.77 − 1.96i)7-s − 2.65i·11-s + (1.59 + 0.920i)13-s + (−0.267 + 0.463i)17-s + (2.57 − 1.48i)19-s − 0.969i·23-s − 3.71·25-s + (2.66 − 1.54i)29-s + (−0.682 + 0.394i)31-s + (−2.00 − 2.22i)35-s + (−1.73 − 3.00i)37-s + (−2.22 + 3.85i)41-s + (−0.849 − 1.47i)43-s + (5.14 − 8.90i)47-s + ⋯
L(s)  = 1  + 0.506·5-s + (−0.670 − 0.742i)7-s − 0.801i·11-s + (0.442 + 0.255i)13-s + (−0.0648 + 0.112i)17-s + (0.590 − 0.340i)19-s − 0.202i·23-s − 0.743·25-s + (0.495 − 0.286i)29-s + (−0.122 + 0.0707i)31-s + (−0.339 − 0.376i)35-s + (−0.284 − 0.493i)37-s + (−0.347 + 0.602i)41-s + (−0.129 − 0.224i)43-s + (0.750 − 1.29i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.270 + 0.962i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.270 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.370638144\)
\(L(\frac12)\) \(\approx\) \(1.370638144\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.77 + 1.96i)T \)
good5 \( 1 - 1.13T + 5T^{2} \)
11 \( 1 + 2.65iT - 11T^{2} \)
13 \( 1 + (-1.59 - 0.920i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.267 - 0.463i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.57 + 1.48i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.969iT - 23T^{2} \)
29 \( 1 + (-2.66 + 1.54i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.682 - 0.394i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.73 + 3.00i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.22 - 3.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.849 + 1.47i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.14 + 8.90i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.9 + 6.87i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.64 + 6.31i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.74 + 1.00i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.20 - 2.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.3iT - 71T^{2} \)
73 \( 1 + (-7.05 - 4.07i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.58 - 4.47i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.56 + 13.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (6.52 + 11.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.44 + 0.832i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.848957211831308003275428301109, −8.061530538009264549590036337517, −7.15956295471682486789284497583, −6.40676088103203255758705268551, −5.81242897087940372615417312294, −4.79631566670544646928365925846, −3.75677981637095959053056382544, −3.07767976846476535116289187865, −1.77796650907160191858445441587, −0.46967258149604069171757230935, 1.43500087380046852807613603891, 2.51605170501802040163055210495, 3.37636752882024724030236789590, 4.47641090799125806367287711059, 5.47811909230944889482364073274, 6.03498865048149126694719038094, 6.85364471096045525425618469405, 7.70715246035266085445599744224, 8.562803228479529947071942224925, 9.431505984048539117955143016790

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