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Properties

Label 2-2268-63.16-c1-0-1
Degree $2$
Conductor $2268$
Sign $-0.557 - 0.829i$
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  − 2.50·5-s + (0.893 − 2.49i)7-s − 1.08·11-s + (−0.457 − 0.792i)13-s + (1.39 + 2.41i)17-s + (0.667 − 1.15i)19-s − 6.12·23-s + 1.28·25-s + (−0.414 + 0.718i)29-s + (−1.30 + 2.26i)31-s + (−2.23 + 6.24i)35-s + (3.14 − 5.44i)37-s + (−0.817 − 1.41i)41-s + (0.795 − 1.37i)43-s + (3.29 + 5.70i)47-s + ⋯
L(s)  = 1  − 1.12·5-s + (0.337 − 0.941i)7-s − 0.327·11-s + (−0.126 − 0.219i)13-s + (0.337 + 0.585i)17-s + (0.153 − 0.265i)19-s − 1.27·23-s + 0.256·25-s + (−0.0769 + 0.133i)29-s + (−0.234 + 0.406i)31-s + (−0.378 + 1.05i)35-s + (0.517 − 0.895i)37-s + (−0.127 − 0.221i)41-s + (0.121 − 0.210i)43-s + (0.480 + 0.832i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.557 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.557 - 0.829i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.557 - 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3536145535\)
\(L(\frac12)\) \(\approx\) \(0.3536145535\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.893 + 2.49i)T \)
good5 \( 1 + 2.50T + 5T^{2} \)
11 \( 1 + 1.08T + 11T^{2} \)
13 \( 1 + (0.457 + 0.792i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.39 - 2.41i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.667 + 1.15i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 6.12T + 23T^{2} \)
29 \( 1 + (0.414 - 0.718i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.30 - 2.26i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.14 + 5.44i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.817 + 1.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.795 + 1.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.29 - 5.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.95 - 6.85i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.53 - 4.39i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.81 - 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.94 - 13.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.64T + 71T^{2} \)
73 \( 1 + (5.03 + 8.72i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.67 - 8.09i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.09 + 7.09i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.36 - 11.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.46 - 14.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.184301058486061358084117536637, −8.336529385705862194116616087680, −7.57597020352712590607224213948, −7.38227031381559482569807555646, −6.18218443232224231890063672909, −5.27398295611804899744268180811, −4.13929443025000585842919315679, −3.89867357316253207051439962970, −2.64845097019233366059713956328, −1.19491050900845098033074990748, 0.13138584745503660364931726659, 1.85268889120328661262852620142, 2.92059546120728324232137384096, 3.86653610905559766872019740591, 4.72735172917881683673597922356, 5.54669712451927627091217807768, 6.39058139559535943535148348600, 7.46183209025267036914177773984, 7.986648175281610226462423467778, 8.570259207149091711719095641510

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