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Properties

Label 2-2268-7.2-c1-0-5
Degree $2$
Conductor $2268$
Sign $0.349 - 0.936i$
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  + (−0.171 + 0.297i)5-s + (−2.41 − 1.08i)7-s + (−2.45 − 4.26i)11-s − 1.94·13-s + (2.07 + 3.59i)17-s + (0.202 − 0.350i)19-s + (−1.37 + 2.38i)23-s + (2.44 + 4.22i)25-s − 3.27·29-s + (1.64 + 2.84i)31-s + (0.736 − 0.533i)35-s + (−3.38 + 5.87i)37-s + 5.62·41-s + 9.92·43-s + (6.60 − 11.4i)47-s + ⋯
L(s)  = 1  + (−0.0768 + 0.133i)5-s + (−0.912 − 0.408i)7-s + (−0.741 − 1.28i)11-s − 0.540·13-s + (0.502 + 0.871i)17-s + (0.0464 − 0.0803i)19-s + (−0.287 + 0.498i)23-s + (0.488 + 0.845i)25-s − 0.608·29-s + (0.295 + 0.511i)31-s + (0.124 − 0.0901i)35-s + (−0.557 + 0.965i)37-s + 0.878·41-s + 1.51·43-s + (0.962 − 1.66i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.349 - 0.936i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.349 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9800346362\)
\(L(\frac12)\) \(\approx\) \(0.9800346362\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.41 + 1.08i)T \)
good5 \( 1 + (0.171 - 0.297i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.45 + 4.26i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.94T + 13T^{2} \)
17 \( 1 + (-2.07 - 3.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.202 + 0.350i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.37 - 2.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.27T + 29T^{2} \)
31 \( 1 + (-1.64 - 2.84i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.38 - 5.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 5.62T + 41T^{2} \)
43 \( 1 - 9.92T + 43T^{2} \)
47 \( 1 + (-6.60 + 11.4i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.38 - 2.39i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.22 - 7.31i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.37 - 9.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.21 - 7.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.08T + 71T^{2} \)
73 \( 1 + (-4.58 - 7.93i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.24 - 3.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.25T + 83T^{2} \)
89 \( 1 + (-3.54 + 6.13i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.62T + 97T^{2} \)
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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

−9.103169436783459914997153841047, −8.458336271257838315252705320666, −7.51916150196394025451603252966, −6.98130954096563195064163331111, −5.85495797455222979578699391733, −5.51236945761620158966514776663, −4.14948409686726977295947856246, −3.37149350639631768777013060743, −2.61752790065664216595499304558, −1.00778777134383706611688229626, 0.39514033451078991229705312751, 2.21771515305325694609415247964, 2.82314443228617830789985784759, 4.07283155290651771369129667315, 4.88769301626250313636313743057, 5.69950620116277357170551456162, 6.57549718646348726271777864895, 7.43280851810216602662375253336, 7.898755361134044379286589080605, 9.169201201346798790135815861691

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