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Properties

Label 2-2268-63.59-c1-0-29
Degree $2$
Conductor $2268$
Sign $-0.723 + 0.690i$
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  + 0.770·5-s + (−0.167 + 2.64i)7-s − 5.40i·11-s + (−4.57 − 2.64i)13-s + (2.85 − 4.94i)17-s + (0.535 − 0.309i)19-s + 6.74i·23-s − 4.40·25-s + (−6.99 + 4.03i)29-s + (−0.502 + 0.290i)31-s + (−0.129 + 2.03i)35-s + (−4.53 − 7.86i)37-s + (−4.29 + 7.44i)41-s + (−1.70 − 2.94i)43-s + (−0.385 + 0.667i)47-s + ⋯
L(s)  = 1  + 0.344·5-s + (−0.0633 + 0.997i)7-s − 1.62i·11-s + (−1.26 − 0.732i)13-s + (0.692 − 1.19i)17-s + (0.122 − 0.0709i)19-s + 1.40i·23-s − 0.881·25-s + (−1.29 + 0.749i)29-s + (−0.0902 + 0.0521i)31-s + (−0.0218 + 0.344i)35-s + (−0.746 − 1.29i)37-s + (−0.670 + 1.16i)41-s + (−0.259 − 0.449i)43-s + (−0.0562 + 0.0973i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.723 + 0.690i$
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: \((2,\ 2268,\ (\ :1/2),\ -0.723 + 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6934336196\)
\(L(\frac12)\) \(\approx\) \(0.6934336196\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.167 - 2.64i)T \)
good5 \( 1 - 0.770T + 5T^{2} \)
11 \( 1 + 5.40iT - 11T^{2} \)
13 \( 1 + (4.57 + 2.64i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.85 + 4.94i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.535 + 0.309i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 6.74iT - 23T^{2} \)
29 \( 1 + (6.99 - 4.03i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.502 - 0.290i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.53 + 7.86i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.29 - 7.44i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.70 + 2.94i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.385 - 0.667i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.63 - 3.83i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.89 + 11.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.57 - 2.06i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.77 + 11.7i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.25iT - 71T^{2} \)
73 \( 1 + (-1.96 - 1.13i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.03 + 6.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.92 + 3.33i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.59 + 4.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.1 - 7.01i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.911196678674799257057845706551, −7.86217505170009094639747105129, −7.38052319052307510445110526672, −6.17509092080592227391660548094, −5.38043054181012750125457708456, −5.22514367564181326173287284004, −3.47871350885706051959360965710, −2.96599916192396395040929919853, −1.82332575867063722240861971157, −0.22006510740895317834105932372, 1.59892729031463957464425515418, 2.36004418525042068496884374316, 3.83734091999189156268935849169, 4.40997123677028139232885843603, 5.25857400712075460403405445209, 6.34291940372229139076061210513, 7.12595839538338708683326114164, 7.55186971157428296292914809981, 8.509099926662270535923224862515, 9.578156408313052438268250802143

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