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Properties

Label 2-2268-7.4-c1-0-4
Degree $2$
Conductor $2268$
Sign $0.167 - 0.985i$
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 + (2.63 + 0.277i)7-s + (−0.301 + 0.522i)11-s − 5.25·13-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 − 7.97·29-s + (−1.57 + 2.72i)31-s + (−3.95 − 8.88i)35-s + (0.00266 + 0.00462i)37-s − 4.01·41-s + 7.32·43-s + (6.10 + 10.5i)47-s + ⋯
L(s)  = 1  + (−0.822 − 1.42i)5-s + (0.994 + 0.104i)7-s + (−0.0909 + 0.157i)11-s − 1.45·13-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 − 1.48·29-s + (−0.282 + 0.490i)31-s + (−0.668 − 1.50i)35-s + (0.000438 + 0.000760i)37-s − 0.627·41-s + 1.11·43-s + (0.891 + 1.54i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.167 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.167 - 0.985i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1621, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.167 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8226309676\)
\(L(\frac12)\) \(\approx\) \(0.8226309676\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 - 0.277i)T \)
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 \( 1 + 5.25T + 13T^{2} \)
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 \( 1 + 7.97T + 29T^{2} \)
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 \( 1 + 4.01T + 41T^{2} \)
43 \( 1 - 7.32T + 43T^{2} \)
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 \( 1 - 12.4T + 83T^{2} \)
89 \( 1 + (-4.09 - 7.09i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−9.070284865060273623932645894949, −8.322485939378483191399814584434, −7.78253247806258707280469935388, −7.22846641949804175154203779724, −5.71947213196051896820184083637, −5.18418845412640632749210186597, −4.39727791246667858318357166152, −3.77120439842200442358276467088, −2.17526712064444449399440035426, −1.22542791293841938259572076911, 0.29619063254712748247880875352, 2.23549534970174117098286041375, 2.87400771410989854285277761504, 3.95173451052721461634016307310, 4.81372927824692254780822308534, 5.59569627422367919314952591260, 6.92664700331957574121547994198, 7.37373655364092900054466506423, 7.66373004045774737215974247448, 8.895170236784106834889277108722

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