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Properties

Label 2-2268-63.25-c1-0-4
Degree $2$
Conductor $2268$
Sign $-0.983 + 0.179i$
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  + (−1.83 + 3.18i)5-s + (−1.55 + 2.14i)7-s + (−0.301 − 0.522i)11-s + (2.62 + 4.55i)13-s + (−2.12 + 3.68i)17-s + (3.68 + 6.38i)19-s + (−0.578 + 1.00i)23-s + (−4.25 − 7.37i)25-s + (3.98 − 6.90i)29-s + 3.15·31-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 − 12.2·47-s + ⋯
L(s)  = 1  + (−0.822 + 1.42i)5-s + (−0.588 + 0.808i)7-s + (−0.0909 − 0.157i)11-s + (0.729 + 1.26i)13-s + (−0.515 + 0.892i)17-s + (0.845 + 1.46i)19-s + (−0.120 + 0.209i)23-s + (−0.851 − 1.47i)25-s + (0.740 − 1.28i)29-s + 0.565·31-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 − 1.78·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.983 + 0.179i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.983 + 0.179i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9566194989\)
\(L(\frac12)\) \(\approx\) \(0.9566194989\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.55 - 2.14i)T \)
good5 \( 1 + (1.83 - 3.18i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.301 + 0.522i)T + (-5.5 + 9.52i)T^{2} \)
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 \( 1 + (0.578 - 1.00i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.98 + 6.90i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.15T + 31T^{2} \)
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 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 + (-4.64 + 8.05i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.61T + 59T^{2} \)
61 \( 1 + 1.93T + 61T^{2} \)
67 \( 1 - 8.63T + 67T^{2} \)
71 \( 1 + 1.13T + 71T^{2} \)
73 \( 1 + (5.33 - 9.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 4.14T + 79T^{2} \)
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

−9.653470055568501065970736217638, −8.352266120632363389819135537549, −8.093847583376830636631217559212, −6.91037589033103010998732228970, −6.39618096520068182246376682325, −5.83556617250832329885520727792, −4.36815210609047078823312557882, −3.60644541083381196755963819612, −2.92249091229794966050821762998, −1.80206303156321272050797598089, 0.38818401211031854177258732723, 1.08127070673455714819844358755, 2.93077207513181295699267788934, 3.69078203509755939946882467262, 4.79953490003998032064056481057, 5.05401091984778861030234354761, 6.33549902341913952702699180854, 7.25224080932381531217820274169, 7.81561314293442684460975449162, 8.745372291699570097498731551125

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