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Properties

Label 2-2268-21.20-c3-0-31
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

\(L(2)\) \(\approx\) \(1.816914307\)
\(L(\frac12)\) \(\approx\) \(1.816914307\)
\(L(\frac{5}{2})\) 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

−8.735178543869244470905559383897, −8.095789986619637810893417138008, −7.38565658950419346115139077107, −6.44587892323350835205459498205, −5.44908864260485833482052753669, −5.07057214612880984876044397480, −3.89056730531002452542826833118, −2.98919410879751592814000356393, −1.97915552608622062308332539273, −0.886266002633938082802148828141, 0.41668424384603053141162111531, 1.78916317772920894077167648724, 2.34556875545415445796208355929, 4.07880435504409403464632635529, 4.24990675364273781625188521589, 5.26559736670009473214671513463, 6.35302856651034201935606568380, 6.95226522678249633390787893402, 7.87997086669797731186961672796, 8.377929018614038075029631321596

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