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Properties

Label 2-2268-63.59-c1-0-3
Degree $2$
Conductor $2268$
Sign $-0.896 - 0.443i$
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  − 2.00·5-s + (1.37 + 2.25i)7-s + 4.80i·11-s + (−0.864 − 0.499i)13-s + (−0.0445 + 0.0772i)17-s + (3.68 − 2.12i)19-s + 0.969i·23-s − 0.985·25-s + (−2.93 + 1.69i)29-s + (4.35 − 2.51i)31-s + (−2.76 − 4.52i)35-s + (−0.0675 − 0.117i)37-s + (−5.62 + 9.73i)41-s + (−3.66 − 6.35i)43-s + (−1.76 + 3.06i)47-s + ⋯
L(s)  = 1  − 0.895·5-s + (0.521 + 0.853i)7-s + 1.44i·11-s + (−0.239 − 0.138i)13-s + (−0.0108 + 0.0187i)17-s + (0.846 − 0.488i)19-s + 0.202i·23-s − 0.197·25-s + (−0.544 + 0.314i)29-s + (0.782 − 0.451i)31-s + (−0.467 − 0.764i)35-s + (−0.0111 − 0.0192i)37-s + (−0.877 + 1.52i)41-s + (−0.559 − 0.968i)43-s + (−0.257 + 0.446i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.896 - 0.443i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.896 - 0.443i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7611886305\)
\(L(\frac12)\) \(\approx\) \(0.7611886305\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.37 - 2.25i)T \)
good5 \( 1 + 2.00T + 5T^{2} \)
11 \( 1 - 4.80iT - 11T^{2} \)
13 \( 1 + (0.864 + 0.499i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.0445 - 0.0772i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.68 + 2.12i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.969iT - 23T^{2} \)
29 \( 1 + (2.93 - 1.69i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.35 + 2.51i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0675 + 0.117i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.62 - 9.73i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.66 + 6.35i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.76 - 3.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.31 + 3.07i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.88 + 8.45i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.2 + 6.51i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.57 - 13.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.56iT - 71T^{2} \)
73 \( 1 + (-3.73 - 2.15i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.94 - 10.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.62 + 2.81i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.06 - 1.84i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.03 - 0.597i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.484292532710705123157832545335, −8.417425815775747066473692259959, −7.82644354269220757802441303319, −7.20882561683692159373229199983, −6.30446478810351828642092652636, −5.11856214220950277355868994747, −4.73176861376605594370487383907, −3.66041517819998892354373819020, −2.61564973190162846119952001134, −1.57316011762921471862629569140, 0.26937820985778064298811175003, 1.46251325817510237779171168786, 3.07184375221516730997998006001, 3.73864156795506588444729827354, 4.56416063216062215799640719386, 5.48577903589355214274027737747, 6.40665620398882937811758454704, 7.35344192700813862663064338052, 7.915911806498332292642905330480, 8.485210520299801066078363947547

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