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Properties

Label 2-2268-63.47-c1-0-13
Degree $2$
Conductor $2268$
Sign $0.805 + 0.592i$
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  − 4.24·5-s + (2 − 1.73i)7-s + (−3 + 1.73i)13-s + (2.12 + 3.67i)17-s + (−1.5 − 0.866i)19-s + 7.34i·23-s + 12.9·25-s + (−6.36 − 3.67i)29-s + (1.5 + 0.866i)31-s + (−8.48 + 7.34i)35-s + (4 − 6.92i)37-s + (−2.12 − 3.67i)41-s + (2.5 − 4.33i)43-s + (2.12 + 3.67i)47-s + (1.00 − 6.92i)49-s + ⋯
L(s)  = 1  − 1.89·5-s + (0.755 − 0.654i)7-s + (−0.832 + 0.480i)13-s + (0.514 + 0.891i)17-s + (−0.344 − 0.198i)19-s + 1.53i·23-s + 2.59·25-s + (−1.18 − 0.682i)29-s + (0.269 + 0.155i)31-s + (−1.43 + 1.24i)35-s + (0.657 − 1.13i)37-s + (−0.331 − 0.573i)41-s + (0.381 − 0.660i)43-s + (0.309 + 0.535i)47-s + (0.142 − 0.989i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.805 + 0.592i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.805 + 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.034189613\)
\(L(\frac12)\) \(\approx\) \(1.034189613\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 + 1.73i)T \)
good5 \( 1 + 4.24T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.12 - 3.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.5 + 0.866i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 7.34iT - 23T^{2} \)
29 \( 1 + (6.36 + 3.67i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.12 + 3.67i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.5 + 4.33i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.12 - 3.67i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.36 + 3.67i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.24 + 7.34i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 2.59i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.6iT - 71T^{2} \)
73 \( 1 + (7.5 - 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.48 - 14.6i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.36 + 11.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-13.5 - 7.79i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.724840045347317673818212239143, −8.002769782678925659773076668974, −7.44243183692161015641401162512, −7.06695229761635234026845314354, −5.68022905925216785617172658015, −4.71728842923964474123798477161, −3.99214441666495014278483914589, −3.51663713933034834562216065417, −1.98223648477032371951545203037, −0.54625573162498444040616972344, 0.78106687361164686192345639806, 2.49534653420934824873479500669, 3.30238671251089986684444082198, 4.45692087635607788470414148003, 4.81917544199796366385994449656, 5.89346708704684201180672945502, 7.17847944712098184336583176679, 7.50760179655974784430552226570, 8.397205395575925508729446671617, 8.703512678003880271503423447934

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