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Properties

Label 2-2268-21.20-c3-0-17
Degree $2$
Conductor $2268$
Sign $-0.463 - 0.885i$
Analytic cond. $133.816$
Root an. cond. $11.5679$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.660·5-s + (16.4 − 8.59i)7-s − 24.7i·11-s + 50.2i·13-s + 67.5·17-s + 62.9i·19-s + 156. i·23-s − 124.·25-s + 149. i·29-s − 161. i·31-s + (10.8 − 5.67i)35-s − 16.2·37-s − 269.·41-s − 376.·43-s − 62.7·47-s + ⋯
L(s)  = 1  + 0.0590·5-s + (0.885 − 0.463i)7-s − 0.677i·11-s + 1.07i·13-s + 0.963·17-s + 0.760i·19-s + 1.42i·23-s − 0.996·25-s + 0.960i·29-s − 0.933i·31-s + (0.0523 − 0.0273i)35-s − 0.0722·37-s − 1.02·41-s − 1.33·43-s − 0.194·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.463 - 0.885i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.463 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.463 - 0.885i$
Analytic conductor: \(133.816\)
Root analytic conductor: \(11.5679\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :3/2),\ -0.463 - 0.885i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.364306673\)
\(L(\frac12)\) \(\approx\) \(1.364306673\)
\(L(\frac{5}{2})\) 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 \( 1 + (-16.4 + 8.59i)T \)
good5 \( 1 - 0.660T + 125T^{2} \)
11 \( 1 + 24.7iT - 1.33e3T^{2} \)
13 \( 1 - 50.2iT - 2.19e3T^{2} \)
17 \( 1 - 67.5T + 4.91e3T^{2} \)
19 \( 1 - 62.9iT - 6.85e3T^{2} \)
23 \( 1 - 156. iT - 1.21e4T^{2} \)
29 \( 1 - 149. iT - 2.43e4T^{2} \)
31 \( 1 + 161. iT - 2.97e4T^{2} \)
37 \( 1 + 16.2T + 5.06e4T^{2} \)
41 \( 1 + 269.T + 6.89e4T^{2} \)
43 \( 1 + 376.T + 7.95e4T^{2} \)
47 \( 1 + 62.7T + 1.03e5T^{2} \)
53 \( 1 - 136. iT - 1.48e5T^{2} \)
59 \( 1 + 717.T + 2.05e5T^{2} \)
61 \( 1 - 27.6iT - 2.26e5T^{2} \)
67 \( 1 - 327.T + 3.00e5T^{2} \)
71 \( 1 - 246. iT - 3.57e5T^{2} \)
73 \( 1 - 261. iT - 3.89e5T^{2} \)
79 \( 1 - 783.T + 4.93e5T^{2} \)
83 \( 1 + 1.19e3T + 5.71e5T^{2} \)
89 \( 1 - 968.T + 7.04e5T^{2} \)
97 \( 1 - 1.27e3iT - 9.12e5T^{2} \)
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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

−8.887215101055111975716465643080, −8.012866060959706884092564726088, −7.56873149412130847321388097464, −6.61055605281827960267693387173, −5.69809925140365623599528485985, −5.03326822270083176103008884565, −3.96996280936444999381557243229, −3.35903412465244057043631569033, −1.87264719384635905978602769862, −1.25105540854325329600120308045, 0.26130865757987085251312325090, 1.50747364333740092588268098904, 2.44807968667352684241048711986, 3.39785827768989740188890291710, 4.61651490272273851537268726745, 5.13474532033635762464113149962, 5.99937098330494714202132532566, 6.90068559577713599153925715283, 7.87647053863063815634133886053, 8.231384261117185383654827300718

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