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Properties

Label 2-2268-63.58-c1-0-20
Degree $2$
Conductor $2268$
Sign $0.888 + 0.458i$
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  + (1.58 + 2.73i)5-s + (−2.5 + 0.866i)7-s + (3.16 − 5.47i)11-s + (−1.58 − 2.73i)17-s + (3.5 − 6.06i)19-s + (−1.58 − 2.73i)23-s + (−2.5 + 4.33i)25-s + (1.58 + 2.73i)29-s + 3·31-s + (−6.32 − 5.47i)35-s + (2 − 3.46i)37-s + (4.74 − 8.21i)41-s + (−2.5 − 4.33i)43-s − 9.48·47-s + (5.5 − 4.33i)49-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)5-s + (−0.944 + 0.327i)7-s + (0.953 − 1.65i)11-s + (−0.383 − 0.664i)17-s + (0.802 − 1.39i)19-s + (−0.329 − 0.571i)23-s + (−0.5 + 0.866i)25-s + (0.293 + 0.508i)29-s + 0.538·31-s + (−1.06 − 0.925i)35-s + (0.328 − 0.569i)37-s + (0.740 − 1.28i)41-s + (−0.381 − 0.660i)43-s − 1.38·47-s + (0.785 − 0.618i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $0.888 + 0.458i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (2053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ 0.888 + 0.458i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.795357807\)
\(L(\frac12)\) \(\approx\) \(1.795357807\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 - 0.866i)T \)
good5 \( 1 + (-1.58 - 2.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.16 + 5.47i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.58 + 2.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.58 + 2.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.58 - 2.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.74 + 8.21i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.48T + 47T^{2} \)
53 \( 1 + (-4.74 - 8.21i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 + 3T + 61T^{2} \)
67 \( 1 - 10T + 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 + (-2.5 - 4.33i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 12T + 79T^{2} \)
83 \( 1 + (-3.16 - 5.47i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.74 - 8.21i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.305096477022791408071718270756, −8.315103778565559545529284397426, −7.06519936923435027013065083193, −6.60069971669457879353718278539, −6.06170338441133871725798984151, −5.20382806900733784698931651739, −3.79304760578202008171899591081, −3.01387356418344255389241164259, −2.44599185217935766994646274390, −0.68237859342056130607649258636, 1.21330507051260742250970811808, 1.95352362471771084177144139153, 3.44736509098059520567381257731, 4.33515889565678688651634725131, 4.99850775498442356973426557244, 6.11954218030747270072485458816, 6.51673170644194946192892428809, 7.62405357596026106334627930216, 8.328398486410653951698214132888, 9.416179123600039970326819362384

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