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Properties

Label 2-2268-63.4-c1-0-30
Degree $2$
Conductor $2268$
Sign $-0.869 - 0.494i$
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  − 3.67·5-s + (−1.07 − 2.41i)7-s − 0.603·11-s + (2.62 − 4.55i)13-s + (2.12 − 3.68i)17-s + (3.68 + 6.38i)19-s − 1.15·23-s + 8.51·25-s + (−3.98 − 6.90i)29-s + (−1.57 − 2.72i)31-s + (3.95 + 8.88i)35-s + (0.00266 + 0.00462i)37-s + (−2.00 + 3.48i)41-s + (−3.66 − 6.34i)43-s + (−6.10 + 10.5i)47-s + ⋯
L(s)  = 1  − 1.64·5-s + (−0.406 − 0.913i)7-s − 0.181·11-s + (0.729 − 1.26i)13-s + (0.515 − 0.892i)17-s + (0.845 + 1.46i)19-s − 0.241·23-s + 1.70·25-s + (−0.740 − 1.28i)29-s + (−0.282 − 0.490i)31-s + (0.668 + 1.50i)35-s + (0.000438 + 0.000760i)37-s + (−0.313 + 0.543i)41-s + (−0.558 − 0.967i)43-s + (−0.891 + 1.54i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.869 - 0.494i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.869 - 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1324139420\)
\(L(\frac12)\) \(\approx\) \(0.1324139420\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.07 + 2.41i)T \)
good5 \( 1 + 3.67T + 5T^{2} \)
11 \( 1 + 0.603T + 11T^{2} \)
13 \( 1 + (-2.62 + 4.55i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.12 + 3.68i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.68 - 6.38i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.15T + 23T^{2} \)
29 \( 1 + (3.98 + 6.90i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.57 + 2.72i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.00266 - 0.00462i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.00 - 3.48i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.66 + 6.34i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.10 - 10.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.64 - 8.05i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.30 + 5.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.969 + 1.67i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.31 + 7.47i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.13T + 71T^{2} \)
73 \( 1 + (5.33 - 9.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.07 - 3.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6.24 - 10.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.09 + 7.09i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.77 - 11.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.093371842794156282017783279050, −7.890921151442728429966120716549, −7.35841570363854869322419560835, −6.26607133724721333068307561214, −5.37293771738031577069316574699, −4.29702489666248333203957295239, −3.58457520902473095390422853534, −3.08506572730029132678134354249, −1.10844480889146587706395569054, −0.05408454524469829932598124193, 1.63631004082488045661576793476, 3.11418830399907727615433744276, 3.63901173229040188770453930016, 4.62480027197511771234997391079, 5.44225114672594489008685664901, 6.57236837326368386181367028196, 7.11861332784075833904617306473, 8.003179757066435866065492031724, 8.768091190583434188372027355891, 9.122224390180530205778954679245

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